3.25 \(\int \frac{x^3}{(a+b \csc (c+d x^2))^2} \, dx\)

Optimal. Leaf size=616 \[ \frac{b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^2 \log \left (a \sin \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac{i b x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{i b^3 x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{i b x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{i b^3 x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{b^2 x^2 \cos \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \sin \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

[Out]

x^4/(4*a^2) - ((I/2)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*
d) + (I*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x
^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^2*Log[1 - (I*a
*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (b^2*Log[b + a*Sin[c + d*x^2]])/(2*a^2
*(a^2 - b^2)*d^2) - (b^3*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)
*d^2) + (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*PolyL
og[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, (I*a*E^(
I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^2*x^2*Cos[c + d*x^2])/(2*a*(a^2 - b^2
)*d*(b + a*Sin[c + d*x^2]))

________________________________________________________________________________________

Rubi [A]  time = 1.20132, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4205, 4191, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^2 \log \left (a \sin \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac{i b x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{i b^3 x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{i b x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{i b^3 x^2 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{b^2 x^2 \cos \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \sin \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Csc[c + d*x^2])^2,x]

[Out]

x^4/(4*a^2) - ((I/2)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*
d) + (I*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x
^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^2*Log[1 - (I*a
*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (b^2*Log[b + a*Sin[c + d*x^2]])/(2*a^2
*(a^2 - b^2)*d^2) - (b^3*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)
*d^2) + (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*PolyL
og[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, (I*a*E^(
I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^2*x^2*Cos[c + d*x^2])/(2*a*(a^2 - b^2
)*d*(b + a*Sin[c + d*x^2]))

Rule 4205

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

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

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

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [B]  time = 15.3154, size = 2446, normalized size = 3.97 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-(b^2*c*Cos[c + d*x^2]) + b^2*(c + d*x^2)*Cos[c + d*x^2])*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2]))/(2*a*(-a
+ b)*(a + b)*d^2*(a + b*Csc[c + d*x^2])^2) + ((-c + d*x^2)*(c + d*x^2)*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])
^2)/(4*a^2*d^2*(a + b*Csc[c + d*x^2])^2) + (Csc[c + d*x^2]^2*(-2*a*b*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a
^2 - b^2]] + 2*(a*b + 2*a^2*c - b^2*c)*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + b*Sqrt[a^2 - b^2]
*Log[Sec[(c + d*x^2)/2]^2] - b*Sqrt[a^2 - b^2]*Log[Sec[(c + d*x^2)/2]^2*(b + a*Sin[c + d*x^2])] - I*(2*a^2 - b
^2)*(Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b - Sqrt[a^2 - b^2]
)] + PolyLog[2, (b*(1 + I*Tan[(c + d*x^2)/2]))/((-I)*a + b + I*Sqrt[a^2 - b^2])]) + I*(2*a^2 - b^2)*(Log[1 + I
*Tan[(c + d*x^2)/2]]*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b + Sqrt[a^2 - b^2])] + PolyLog[2
, (b*(1 + I*Tan[(c + d*x^2)/2]))/(b - I*(a + Sqrt[a^2 - b^2]))]) - I*(2*a^2 - b^2)*(Log[1 - I*Tan[(c + d*x^2)/
2]]*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b + Sqrt[a^2 - b^2])] + PolyLog[2, -((b*(I + Tan[(
c + d*x^2)/2]))/(a - I*b + Sqrt[a^2 - b^2]))]) + I*(2*a^2 - b^2)*(Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(a - Sqrt[
a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b - Sqrt[a^2 - b^2])] + PolyLog[2, (b*(I + Tan[(c + d*x^2)/2]))/(-a
+ I*b + Sqrt[a^2 - b^2])]))*(b + a*Sin[c + d*x^2])^2*((2*b*c)/((a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) - (b^3*c)
/(a^2*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) - (2*b*(c + d*x^2))/((a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) + (b^3*
(c + d*x^2))/(a^2*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) + (b^2*Cos[c + d*x^2])/(a*(a^2 - b^2)*d*(b + a*Sin[c +
d*x^2]))))/(2*d*(a + b*Csc[c + d*x^2])^2*(b*Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2] - (b*Sqrt[a^2 - b^2]*Cos[(c +
d*x^2)/2]^2*(a*Cos[c + d*x^2]*Sec[(c + d*x^2)/2]^2 + Sec[(c + d*x^2)/2]^2*(b + a*Sin[c + d*x^2])*Tan[(c + d*x^
2)/2]))/(b + a*Sin[c + d*x^2]) - (a*b^2*Sec[(c + d*x^2)/2]^2)/(Sqrt[a^2 - b^2]*(1 - (a + b*Tan[(c + d*x^2)/2])
^2/(a^2 - b^2))) + (b*(a*b + 2*a^2*c - b^2*c)*Sec[(c + d*x^2)/2]^2)/(Sqrt[a^2 - b^2]*(1 - (a + b*Tan[(c + d*x^
2)/2])^2/(a^2 - b^2))) + I*(2*a^2 - b^2)*(((-I/2)*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b -
Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 - I*Tan[(c + d*x^2)/2]) - (Log[1 - (b*(I + Tan[(c + d*x^2)/2]))/(-a
+ I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(2*(I + Tan[(c + d*x^2)/2])) + (b*Log[1 - I*Tan[(c + d*x^2)/2
]]*Sec[(c + d*x^2)/2]^2)/(2*(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]))) - I*(2*a^2 - b^2)*(((-I/2)*Log[1 -
(b*(1 + I*Tan[(c + d*x^2)/2]))/((-I)*a + b + I*Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/
2]) + ((I/2)*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b - Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^
2)/(1 + I*Tan[(c + d*x^2)/2]) + (b*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a - Sqrt[a^2 - b^2]
+ b*Tan[(c + d*x^2)/2]))) - I*(2*a^2 - b^2)*(((-I/2)*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*
b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 - I*Tan[(c + d*x^2)/2]) - (Log[1 + (b*(I + Tan[(c + d*x^2)/2]))
/(a - I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(2*(I + Tan[(c + d*x^2)/2])) + (b*Log[1 - I*Tan[(c + d*x^2
)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]))) + I*(2*a^2 - b^2)*(((-I/2)*Log[1
- (b*(1 + I*Tan[(c + d*x^2)/2]))/(b - I*(a + Sqrt[a^2 - b^2]))]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/
2]) + ((I/2)*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^
2)/(1 + I*Tan[(c + d*x^2)/2]) + (b*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a + Sqrt[a^2 - b^2]
+ b*Tan[(c + d*x^2)/2])))))

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Maple [F]  time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( a+b\csc \left ( d{x}^{2}+c \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*csc(d*x^2+c))^2,x)

[Out]

int(x^3/(a+b*csc(d*x^2+c))^2,x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.07458, size = 4188, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*x^4*sin(d*x^2 + c) + (a^4*b - 2*a^2*b^3 + b^5)*d^2*x^4 - 2*(a^3*b^2 - a*b^4
)*d*x^2*cos(d*x^2 + c) + (2*I*a^3*b^2 - I*a*b^4 + (2*I*a^4*b - I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2
)*dilog(-1/2*(2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2
- b^2)/a^2) + 2*a)/a + 1) + (-2*I*a^3*b^2 + I*a*b^4 + (-2*I*a^4*b + I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2
)/a^2)*dilog(-1/2*(2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(
(a^2 - b^2)/a^2) + 2*a)/a + 1) + (-2*I*a^3*b^2 + I*a*b^4 + (-2*I*a^4*b + I*a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2
- b^2)/a^2)*dilog(-1/2*(-2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))
*sqrt((a^2 - b^2)/a^2) + 2*a)/a + 1) + (2*I*a^3*b^2 - I*a*b^4 + (2*I*a^4*b - I*a^2*b^3)*sin(d*x^2 + c))*sqrt((
a^2 - b^2)/a^2)*dilog(-1/2*(-2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 +
c))*sqrt((a^2 - b^2)/a^2) + 2*a)/a + 1) + ((2*a^3*b^2 - a*b^4)*d*x^2 + (2*a^3*b^2 - a*b^4)*c + ((2*a^4*b - a^
2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*log(1/2*(2*I*b*cos(d*x^2 + c) + 2*
b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) + 2*a)/a) - ((2*a^3*b^2 - a
*b^4)*d*x^2 + (2*a^3*b^2 - a*b^4)*c + ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt
((a^2 - b^2)/a^2)*log(1/2*(2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c
))*sqrt((a^2 - b^2)/a^2) + 2*a)/a) + ((2*a^3*b^2 - a*b^4)*d*x^2 + (2*a^3*b^2 - a*b^4)*c + ((2*a^4*b - a^2*b^3)
*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*log(1/2*(-2*I*b*cos(d*x^2 + c) + 2*b*sin
(d*x^2 + c) + 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) + 2*a)/a) - ((2*a^3*b^2 - a*b^4)
*d*x^2 + (2*a^3*b^2 - a*b^4)*c + ((2*a^4*b - a^2*b^3)*d*x^2 + (2*a^4*b - a^2*b^3)*c)*sin(d*x^2 + c))*sqrt((a^2
- b^2)/a^2)*log(1/2*(-2*I*b*cos(d*x^2 + c) + 2*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*s
qrt((a^2 - b^2)/a^2) + 2*a)/a) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*sin(d*x^2 + c) - ((2*a^4*b - a^2*b^3)*c*si
n(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c)*sqrt((a^2 - b^2)/a^2))*log(2*a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 + c) + 2
*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*sin(d*x^2 + c) - ((2*a^4*b - a^2*b^3)*c
*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c)*sqrt((a^2 - b^2)/a^2))*log(2*a*cos(d*x^2 + c) - 2*I*a*sin(d*x^2 + c)
+ 2*a*sqrt((a^2 - b^2)/a^2) - 2*I*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*sin(d*x^2 + c) + ((2*a^4*b - a^2*b^3
)*c*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c)*sqrt((a^2 - b^2)/a^2))*log(-2*a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 +
c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*sin(d*x^2 + c) + ((2*a^4*b - a^2
*b^3)*c*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c)*sqrt((a^2 - b^2)/a^2))*log(-2*a*cos(d*x^2 + c) - 2*I*a*sin(d*x
^2 + c) + 2*a*sqrt((a^2 - b^2)/a^2) - 2*I*b))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*sin(d*x^2 + c) + (a^6*b - 2*a^4
*b^3 + a^2*b^5)*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a + b \csc{\left (c + d x^{2} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a + b*csc(c + d*x**2))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/(b*csc(d*x^2 + c) + a)^2, x)