\(\int \frac {x^2 (a+b \csc ^{-1}(c x))}{d+e x^2} \, dx\) [98]

Optimal result

Integrand size = 21, antiderivative size = 565 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5349, 4817, 4723, 272, 65, 214, 4757, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e^{3/2}}+\frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^{3/2}} \]

[In]

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

[Out]

(x*(a + b*ArcCsc[c*x]))/e + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e) - (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])])/(2*e^(3/2)) + (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])])/(2*e^(3/2)) - (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])])/(2*e^(3/2)) + (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])])/(2*e^(3/2)) - ((I/2)*b*Sqrt[-d]*P
olyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLo
g[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(3/2) - ((I/2)*b*Sqrt[-d]*PolyLog[2, ((-
I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqr
t[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(3/2)

Rule 65

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 214

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

Rule 272

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 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 4723

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 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 5349

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*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{x^2 \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {a+b \arcsin \left (\frac {x}{c}\right )}{e x^2}-\frac {d \left (a+b \arcsin \left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e}+\frac {d \text {Subst}\left (\int \left (\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \arcsin \left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^{3/2}}+\frac {d \text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^{3/2}}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{e} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c e}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}}+\frac {d \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c 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^{3/2}}+\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^{3/2}}-\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^{3/2}}+\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^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c 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^{3/2}}+\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^{3/2}}-\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^{3/2}}+\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^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}-\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}}+\frac {\left (i b \sqrt {-d}\right ) \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 e^{3/2}} \\ & = \frac {x \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c 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^{3/2}}+\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^{3/2}}-\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^{3/2}}+\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^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}}+\frac {i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1260\) vs. \(2(565)=1130\).

Time = 1.82 (sec) , antiderivative size = 1260, normalized size of antiderivative = 2.23 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {i \left (-4 i a c \sqrt {e} x-4 i b c \sqrt {e} x \csc ^{-1}(c x)+4 i a c \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 i b c \sqrt {d} \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )-8 i b c \sqrt {d} \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )+b c \sqrt {d} \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b c \sqrt {d} \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 b c \sqrt {d} \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-b c \sqrt {d} \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b c \sqrt {d} \csc ^{-1}(c x) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-4 b c \sqrt {d} \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-b c \sqrt {d} \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 b c \sqrt {d} \csc ^{-1}(c x) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+4 b c \sqrt {d} \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+b c \sqrt {d} \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 b c \sqrt {d} \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-4 b c \sqrt {d} \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+b c \sqrt {d} \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-b c \sqrt {d} \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )-4 i b \sqrt {e} \log \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )+4 i b \sqrt {e} \log \left (\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )+2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b c \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{4 c e^{3/2}} \]

[In]

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

[Out]

((I/4)*((-4*I)*a*c*Sqrt[e]*x - (4*I)*b*c*Sqrt[e]*x*ArcCsc[c*x] + (4*I)*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
+ (8*I)*b*c*Sqrt[d]*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]] - (8*I)*b*c*Sqrt[d]*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]] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt
[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*
d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*c*Sqrt[d]*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]))] - b*c*Sqrt[d]*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d
+ e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[
d]*E^(I*ArcCsc[c*x]))] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + S
qrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*c*Sqrt[d]*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]
*E^(I*ArcCsc[c*x]))] + 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*
x]))] + 4*b*c*Sqrt[d]*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]))] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])
)] - 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*c*Sqrt
[d]*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*ArcC
sc[c*x]))] + b*c*Sqrt[d]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*c*Sqrt[d]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - (4*I)
*b*Sqrt[e]*Log[Cos[ArcCsc[c*x]/2]] + (4*I)*b*Sqrt[e]*Log[Sin[ArcCsc[c*x]/2]] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (S
qrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*c*Sqrt[d]*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d
+ e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*c*Sqrt[d]*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*
E^(I*ArcCsc[c*x])))] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]
))/(c*e^(3/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 44.19 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.73

method	result	size
parts	\(\frac {a x}{e}-\frac {a d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arccsc}\left (c x \right ) x}{e}-\frac {c^{2} \ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}+\frac {c^{4} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\)	\(415\)
derivativedivides	\(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\)	\(417\)
default	\(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e \sqrt {d e}}+b \,c^{2} \left (\frac {c x \,\operatorname {arccsc}\left (c x \right )}{e}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d -c^{2} d -4 e \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}-\frac {\ln \left (-1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +4 \textit {\_R1}^{2} e -c^{2} d \right ) \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{8 e^{2}}\right )}{c^{3}}\)	\(417\)

[In]

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

[Out]

a/e*x-a*d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c^3*(c^3*arccsc(c*x)/e*x-c^2/e*ln(-1+I/c/x+(1-1/c^2/x^2)^(1/
2))+c^2/e*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/8*c^4/e^2*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=RootO
f(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))+1/8*c^4/e^2*d*sum((_R1^2*c^2*d+4*_R1^2*e-c^2*d)/_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)))

Fricas [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{e x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]

[In]

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

[Out]

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