3.107 \(\int \frac {x^4 (a+b \csc ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=803 \[ \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}} \]

[Out]

x*(a+b*arccsc(c*x))/e^2+b*arctanh((1-1/c^2/x^2)^(1/2))/c/e^2-3/4*(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)^(1/2)/e^(5/2)+3/4*(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)^(1/2)/e^(5/2)-3/4*(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)^(1/2)/e^(5/2)+3/4*(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)^(1/2)/e^(5/2)-3/4*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)^(1/2)/e^(5/2)+3/4*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)^(1/2)/e^(5/2)-3/4*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)^(1/2)/e^(5/2)+3/4*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)^(1/2)/e^(5/2)-1/4*d*(a+b*arccsc(c*x))/e^2/(-d/x+(
-d)^(1/2)*e^(1/2))+1/4*d*(a+b*arccsc(c*x))/e^2/(d/x+(-d)^(1/2)*e^(1/2))+1/4*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^(1/2)/e^2/(c^2*d+e)^(1/2)+1/4*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^(1/2)/e^2/(c^2*d+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.40, antiderivative size = 803, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5241, 4733, 4627, 266, 63, 208, 4667, 4743, 725, 206, 4741, 4519, 2190, 2279, 2391} \[ \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right ) \left (a+b \csc ^{-1}(c x)\right )}{4 e^{5/2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{4 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 e^2 \sqrt {d c^2+e}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]

[Out]

-(d*(a + b*ArcCsc[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) + (d*(a + b*ArcCsc[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] +
d/x)) + (x*(a + b*ArcCsc[c*x]))/e^2 + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e^2) + (b*Sqrt[d]*ArcTanh[(c^2*d
- (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) + (b*Sqrt[
d]*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*
d + e]) - (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])
])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*
d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sq
rt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[
e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sq
rt[e] - Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e
] - Sqrt[c^2*d + e])])/e^(5/2) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e]
+ Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + S
qrt[c^2*d + e])])/e^(5/2)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rule 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 725

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 4519

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 4627

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

Rule 4667

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 4733

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 4741

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 4743

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 5241

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

Rubi steps

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

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Mathematica [B]  time = 6.17, size = 1634, normalized size = 2.03 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]

[Out]

(a*x)/e^2 + (a*d*x)/(2*e^2*(d + e*x^2)) - (3*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(5/2)) + b*(-1/4*(d*(
-(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]))/e^2 - (d*(-(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))/(Sq
rt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(4*e^2) + (3*Sqrt[d]*(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])]))/(32*e^(5/2)) - (3*Sqrt[d]*(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)*Ar
cCsc[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 - (Sq
rt[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]
)]))/(32*e^(5/2)) + ((ArcCsc[c*x]*Cot[ArcCsc[c*x]/2])/2 + Log[Cos[ArcCsc[c*x]/2]] - Log[Sin[ArcCsc[c*x]/2]] +
(ArcCsc[c*x]*Tan[ArcCsc[c*x]/2])/2)/(c*e^2))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arccsc}\left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 19.15, size = 1888, normalized size = 2.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/e^2+1/2*c^2*a/e^2*d*x/(c^2*e*x^2+c^2*d)-3/2*a/e^2*d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+c^2*b*x^3*arccsc(c
*x)/e/(c^2*e*x^2+c^2*d)+3/2*c^2*b*x*arccsc(c*x)/e^2/(c^2*e*x^2+c^2*d)*d+1/c*b/e^2*ln(1+I/c/x+(1-1/c^2/x^2)^(1/
2))+1/2/c^2*b*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(
e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e^2/d+1/c^4*b*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x
+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e^2/d^2*(e*(c^2*d+e))^(1/2)+1/c^4*b*(-(c^2
*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2
*e)*d)^(1/2))/e/d^2-1/2/c^2*b*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/
2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e^2/(c^2*d+e)/d*(e*(c^2*d+e))^(1/2)-1/c^2*b*(-(c^2*d-2*(e*(c
^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/
2))/e/(c^2*d+e)/d-1/c^4*b*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/
((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e/(c^2*d+e)/d^2*(e*(c^2*d+e))^(1/2)-1/c^4*b*(-(c^2*d-2*(e*(c^2*d
+e))^(1/2)+2*e)*d)^(1/2)*arctan(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/
(c^2*d+e)/d^2+1/2/c^2*b*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((
c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e^2/d-1/c^4*b*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c
*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e^2/d^2*(e*(c^2*d+e))^(1/2)+1/c^4*
b*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^
(1/2)+2*e)*d)^(1/2))/e/d^2+1/2/c^2*b*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x
^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e^2/(c^2*d+e)/d*(e*(c^2*d+e))^(1/2)-1/c^2*b*((c^2*d+2*
(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d
)^(1/2))/e/(c^2*d+e)/d+1/c^4*b*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x^2)^(1
/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e/(c^2*d+e)/d^2*(e*(c^2*d+e))^(1/2)-1/c^4*b*((c^2*d+2*(e*(c^
2*d+e))^(1/2)+2*e)*d)^(1/2)*arctanh(c*d*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2
))/(c^2*d+e)/d^2-1/c*b/e^2*ln(I/c/x+(1-1/c^2/x^2)^(1/2)-1)-3/16*c*b/e^3*d*sum((_R1^2*c^2*d-c^2*d-4*e)/_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*c*b/e^3*d*sum((_R1^2*c^2*d+4*_R1^2*e-c^2*d)/_R
1/(_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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {d x}{e^{3} x^{2} + d e^{2}} - \frac {3 \, d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {2 \, x}{e^{2}}\right )} + b \int \frac {x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^2,x)

[Out]

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acsc(c*x))/(e*x**2+d)**2,x)

[Out]

Timed out

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