3.2.17 \(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x^2)^3} \, dx\) [117]

Optimal. Leaf size=1134 \[ -\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]

[Out]

-1/16*b*e*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d+e
)^(3/2)-1/16*b*e*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(
c^2*d+e)^(3/2)-3/16*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)
))/(-d)^(5/2)/e^(1/2)+3/16*(a+b*arccsc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e
)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(
c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)+3/16*(a+b*arccsc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^
(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)+3/16*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1
/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/
2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2
)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)+3/16*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-
(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)+1/16*(a+b*arccsc(c*x))*e^(1/2)/(-d)^(3/2)/(-d/x+(-d)^(1/2)*e^(1/2))^2-5/1
6*(a+b*arccsc(c*x))/d^2/(-d/x+(-d)^(1/2)*e^(1/2))-1/16*(a+b*arccsc(c*x))*e^(1/2)/(-d)^(3/2)/(d/x+(-d)^(1/2)*e^
(1/2))^2+5/16*(a+b*arccsc(c*x))/d^2/(d/x+(-d)^(1/2)*e^(1/2))+5/16*b*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^(
1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)+5/16*b*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/
c/d^(1/2)/(c^2*d+e)^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)-1/16*b*c*e^(1/2)*(1-1/c^2/x^2)^(1/2)/(-
d)^(3/2)/(c^2*d+e)/(-d/x+(-d)^(1/2)*e^(1/2))-1/16*b*c*e^(1/2)*(1-1/c^2/x^2)^(1/2)/(-d)^(3/2)/(c^2*d+e)/(d/x+(-
d)^(1/2)*e^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 3.54, antiderivative size = 1134, normalized size of antiderivative = 1.00, number of steps used = 81, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5339, 4817, 4757, 4827, 745, 739, 212, 4825, 4615, 2221, 2317, 2438} \begin {gather*} -\frac {b \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {d c^2+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (d c^2+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {d c^2+e}}-\frac {b e \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (d c^2+e\right )^{3/2}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/(d + e*x^2)^3,x]

[Out]

-1/16*(b*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)])/((-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*c*Sqrt[e]*Sqr
t[1 - 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcCsc[c*x]))/(16*(
-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcCsc[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(
a + b*ArcCsc[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcCsc[c*x]))/(16*d^2*(Sqrt[-d]*Sqr
t[e] + d/x)) - (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])
/(16*d^(5/2)*(c^2*d + e)^(3/2)) + (5*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[
1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) - (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt
[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d + e)^(3/2)) + (5*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])
/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) - (3*(a + b*ArcCsc[c*x])*
Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*Arc
Csc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*
(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt
[e]) + (3*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^
(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((
-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(
(-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])
])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]
)])/((-d)^(5/2)*Sqrt[e])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4757

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4817

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 4825

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cos[x]/(
c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4827

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(
n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5339

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)
^p*((a + b*ArcSin[x/c])^n/x^(2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && Integer
Q[p]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=-\text {Subst}\left (\int \frac {x^4 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {e^2 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^3}-\frac {2 e \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}+\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{d^2}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \left (-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-d x\right )^3}-\frac {3 d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d^3 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}+d x\right )^3}-\frac {3 d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {3 d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}\\ &=\frac {3 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 d}+\frac {3 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 d}+\frac {3 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{8 d}-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{d}-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 d^2 \sqrt {e}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {-d}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {-d}}\\ &=\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2}+\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2}+\frac {3 \text {Subst}\left (\int \left (-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{8 d}-\frac {\text {Subst}\left (\int \left (-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c (-d)^{3/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c (-d)^{3/2}}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 c d^2}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 c d^2}-\frac {b \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 c d^2}+\frac {b \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 c d^2}-\frac {3 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 d^2 \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{16 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 d^2 \sqrt {e}}-\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2 \left (c^2 d+e\right )}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2 \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {(b e) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 c d^2 \left (c^2 d+e\right )}-\frac {(b e) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 c d^2 \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}-\frac {3 \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{16 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}+\frac {\text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{2 d^2 \sqrt {e}}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{2 (-d)^{5/2} \sqrt {e}}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{2 (-d)^{5/2} \sqrt {e}}\\ &=-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {5 b \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d+e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}\\ \end {align*}

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Mathematica [A]
time = 6.03, size = 2060, normalized size = 1.82 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^3,x]

[Out]

(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]
) + b*((-3*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]
*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I
*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Arc
Sin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c
^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) + ((I/16)*
((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]
*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d +
e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[
e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d +
e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) +
(I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/
(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(Pi^2 - 4*Pi*Ar
cCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + S
qrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d
]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] +
(16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*
ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*
x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*
Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[
1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])
/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] +
4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(128*d^(5/2)*Sqrt[e]) + (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 -
32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])
/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*A
rcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqr
t[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log
[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^
2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (S
qrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] -
(4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))
] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x
])]))/(128*d^(5/2)*Sqrt[e]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 13.70, size = 3181, normalized size = 2.81

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3181\)
default	\(\text {Expression too large to display}\)	\(3181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(3/8*a*c^3/d^2*x/(c^2*e*x^2+c^2*d)+b/c^3*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan((I/c/x+(1-1
/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)*(e*(c^2*d+e))^(1/2)-7/4*b/c*(
-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^
(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)^2*(e*(c^2*d+e))^(1/2)-b/c^3*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^
2*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)^2*(e*(c^2
*d+e))^(1/2)-b/c^3*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2
*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)*(e*(c^2*d+e))^(1/2)-1/8*b*c^6*x^4/(c^2*e*x^2+c^2*d)^2/(c
^2*d+e)/d^2*((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+5/8*b*c^5*x/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)/d*e*arccsc(c*x)+3/8*b*c^
5*x^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)/d^2*e^2*arccsc(c*x)-1/8*b*c^6*x^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)/d*((c^2*x^
2-1)/c^2/x^2)^(1/2)*e-5/4*b*c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*
c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)^2*e/d^3-5/4*b*c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)
*d)^(1/2)*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)^2*e/d
^3-5/8*b*c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(
c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)^2/d^3*(e*(c^2*d+e))^(1/2)+5/8*b*c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*
d)^(1/2)*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)^2/d^3*
(e*(c^2*d+e))^(1/2)+b/c^3*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*
c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)-9/4*b/c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(
1/2)*e^2*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)^2-
b/c^3*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^3*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^
2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)^2+5/4*b/c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan((I/c/
x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2*d+e))^(1/2)+3/8
*b*c^7*x^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)/d*e*arccsc(c*x)+7/4*b/c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*
e*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)+b/c^3*(-(
c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^
(1/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)-9/4*b/c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan((I/c/x+(1-1
/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)^2-b/c^3*(-(c^2*d-2*(e*(c^2*d+
e))^(1/2)+2*e)*d)^(1/2)*e^3*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2
))/d^5/(c^2*d+e)^2+7/4*b/c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c
*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)-5/4*b/c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1
/2)*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2
*d+e))^(1/2)+5/8*b*c*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2
*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)/d^3+5/8*b*c*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arc
tan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)/d^3+7/4*b/c*((c^2*
d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+
2*e)*d)^(1/2))/d^4/(c^2*d+e)^2*(e*(c^2*d+e))^(1/2)+b/c^3*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arcta
nh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)^2*(e*(c^2*d+e))^
(1/2)+3/8*a*c/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-3/16*b*c^4/d/(c^2*d+e)*sum(1/_R1/(_R1^2*c^2*d-c^2*d-2*e)
*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf
(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-3/16*b*c^4/d/(c^2*d+e)*sum(_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*
ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*
c^2*d-4*e)*_Z^2+c^2*d))+5/8*b*c^7*x/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccsc(c*x)-3/16*b*c^2/d^2/(c^2*d+e)*e*sum(1
/_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/
x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-3/16*b*c^2/d^2/(c^2*d+e)*e*sum(_R1/(_R1^2*
c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/
_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))+1/4*a*c^5*x/d/(c^2*e*x^2+c^2*d)^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/8*a*((3*x^3*e + 5*d*x)/(d^2*x^4*e^2 + 2*d^3*x^2*e + d^4) + 3*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(5/2)) + b
*integrate(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*arccsc(c*x) + a)/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(sageVARx)]s
ym2poly/r2sym(

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/(d + e*x^2)^3,x)

[Out]

int((a + b*asin(1/(c*x)))/(d + e*x^2)^3, x)

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