3.23 \(\int \frac{x^5}{(a+b \csc (c+d x^2))^2} \, dx\)

Optimal. Leaf size=1124 \[ \text{result too large to display} \]

[Out]

((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b - Sqrt[a^2 -
b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b
^2)*d^2) - ((I/2)*b^3*x^4*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^4*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^4*
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^4*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*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))
/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2
- b^2]))])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(
-a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*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*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2)
- (2*b*x^2*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) - (I*b^3*Po
lyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, (
I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, (I*a*E^(I*(c +
d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(
b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (b^2*x^4*Cos[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Sin[c
+ d*x^2]))

________________________________________________________________________________________

Rubi [A]  time = 2.378, antiderivative size = 1124, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4205, 4191, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4521, 2279, 2391} \[ \frac{x^6}{6 a^2}+\frac{i b \log \left (1-\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{i b^3 \log \left (1-\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{i b \log \left (1-\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{i b^3 \log \left (1-\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}-\frac{b^2 \cos \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (d x^2+c\right )\right )}+\frac{b^2 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{i b-\sqrt{a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{i b+\sqrt{a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b \text{PolyLog}\left (2,\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{2 b \text{PolyLog}\left (2,\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{PolyLog}\left (2,\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{i b^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{2 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}-\frac{i b^3 \text{PolyLog}\left (3,\frac{i a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac{2 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}+\frac{i b^3 \text{PolyLog}\left (3,\frac{i a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Csc[c + d*x^2])^2,x]

[Out]

((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b - Sqrt[a^2 -
b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b
^2)*d^2) - ((I/2)*b^3*x^4*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^4*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^4*
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^4*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*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))
/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2
- b^2]))])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(
-a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*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*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2)
- (2*b*x^2*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) - (I*b^3*Po
lyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, (
I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, (I*a*E^(I*(c +
d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(
b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (b^2*x^4*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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4521

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] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

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]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \sin (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^6}{6 a^2}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{(b+a \sin (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac{x^6}{6 a^2}-\frac{b^2 x^4 \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^2}{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^2}{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{x \cos (c+d x)}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}-\frac{b^2 x^4 \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^2}{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}{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}{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{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b-\sqrt{a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b+\sqrt{a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{i b x^4 \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^4 \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 x^4 \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}{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}{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{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{i (c+d x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{i (c+d x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{(2 i b) \operatorname{Subst}\left (\int x \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{(2 i b) \operatorname{Subst}\left (\int x \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^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \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^4 \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^4 \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^4 \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{2 b x^2 \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{2 b x^2 \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^4 \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^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{i b-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\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^2}+\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\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^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \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 \left (-a^2+b^2\right )^{3/2} d}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \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 \left (-a^2+b^2\right )^{3/2} d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \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^4 \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^4 \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^4 \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^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \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 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \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^4 \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 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b-\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^3}-\frac{(2 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b+\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^3}+\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (\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 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (\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 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \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^4 \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^4 \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^4 \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^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \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 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \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{2 i b \text{Li}_3\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^3}-\frac{2 i b \text{Li}_3\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^3}-\frac{b^2 x^4 \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{\text{Li}_2\left (\frac{i a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \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^4 \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^4 \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^4 \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^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b-\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{i b+\sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \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 x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \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{i b^3 \text{Li}_3\left (\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{2 i b \text{Li}_3\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^3}+\frac{i b^3 \text{Li}_3\left (\frac{i a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{2 i b \text{Li}_3\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^3}-\frac{b^2 x^4 \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 [A]  time = 9.35085, size = 2033, normalized size = 1.81 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*Csc[c + d*x^2])^2,x]

[Out]

(Csc[c/2]*Csc[c + d*x^2]^2*Sec[c/2]*(-(b^3*x^4*Cos[c]) - a*b^2*x^4*Sin[d*x^2])*(b + a*Sin[c + d*x^2]))/(4*a^2*
(-a + b)*(a + b)*d*(a + b*Csc[c + d*x^2])^2) + (x^6*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])^2)/(6*a^2*(a + b*C
sc[c + d*x^2])^2) + (b*E^((2*I)*c)*Csc[c + d*x^2]^2*((-2*I)*b*d^2*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^
4 - 2*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^
((2*I)*c)])] + 2*b*d*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c
) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqr
t[(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 -
b^2)*E^((2*I)*c)])] - 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2
)*E^((2*I)*c)])] + b^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^(
(2*I)*c)])] - 2*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2
- b^2)*E^((2*I)*c)])] + 2*b*d*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(
I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(
I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + b^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sq
rt[(a^2 - b^2)*E^((2*I)*c)])] + 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[
(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2
- b^2)*E^((2*I)*c)])] - (2*I)*(-1 + E^((2*I)*c))*(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*
d*E^(I*c)*x^2)*PolyLog[2, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*(-1
+ E^((2*I)*c))*(-(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]) - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a
*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 4*a^2*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*
c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*b^2*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)
))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 4*a^2*E^((3*I)*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*
E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*b^2*E^((3*I)*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c
) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 4*a^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt
[(a^2 - b^2)*E^((2*I)*c)]))] + 2*b^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b
^2)*E^((2*I)*c)]))] + 4*a^2*E^((3*I)*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E
^((2*I)*c)]))] - 2*b^2*E^((3*I)*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*
I)*c)]))])*(b + a*Sin[c + d*x^2])^2)/(2*a^2*d^3*((a^2 - b^2)*E^((2*I)*c))^(3/2)*(-1 + E^((2*I)*c))*(a + b*Csc[
c + d*x^2])^2)

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Maple [F]  time = 0.466, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \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^5/(a+b*csc(d*x^2+c))^2,x)

[Out]

int(x^5/(a+b*csc(d*x^2+c))^2,x)

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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^5/(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 1.13322, size = 6602, normalized size = 5.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")

[Out]

1/24*(4*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6*sin(d*x^2 + c) + 4*(a^4*b - 2*a^2*b^3 + b^5)*d^3*x^6 - 12*(a^3*b^2 -
a*b^4)*d^2*x^4*cos(d*x^2 + c) - 12*(2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/
a^2)*polylog(3, 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))*sqr
t((a^2 - b^2)/a^2))/a) + 12*(2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*pol
ylog(3, 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))/a) - 12*(2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*polylog(3,
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))/a) + 12*(2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2)*polylog(3, 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))/a)
+ (12*I*a^2*b^3 - 12*I*b^5 + (12*I*a^3*b^2 - 12*I*a*b^4)*sin(d*x^2 + c) + 2*(6*I*(2*a^4*b - a^2*b^3)*d*x^2*si
n(d*x^2 + c) + 6*I*(2*a^3*b^2 - a*b^4)*d*x^2)*sqrt((a^2 - b^2)/a^2))*dilog(-1/2*(2*I*b*cos(d*x^2 + c) + 2*b*si
n(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) + (12*I*a^2*b^3 -
12*I*b^5 + (12*I*a^3*b^2 - 12*I*a*b^4)*sin(d*x^2 + c) + 2*(-6*I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) - 6*
I*(2*a^3*b^2 - a*b^4)*d*x^2)*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) + (-12*I*a^2*b^3 + 12*I*b^5 + (-12
*I*a^3*b^2 + 12*I*a*b^4)*sin(d*x^2 + c) + 2*(-6*I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) - 6*I*(2*a^3*b^2 -
a*b^4)*d*x^2)*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) + (-12*I*a^2*b^3 + 12*I*b^5 + (-12*I*a^3*b^2 + 1
2*I*a*b^4)*sin(d*x^2 + c) + 2*(6*I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) + 6*I*(2*a^3*b^2 - a*b^4)*d*x^2)*s
qrt((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) - 6*(2*(a^3*b^2 - a*b^4)*c*sin(d*x^2 + c) + 2*(a^2*b^3 - b^5)*c
- ((2*a^4*b - a^2*b^3)*c^2*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c^2)*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) - 6*(2*(a^3*b^2 - a*b^4)*c*sin(d*x^2 + c) +
2*(a^2*b^3 - b^5)*c - ((2*a^4*b - a^2*b^3)*c^2*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c^2)*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) - 6*(2*(a^3*b^2 - a*b^4)*
c*sin(d*x^2 + c) + 2*(a^2*b^3 - b^5)*c + ((2*a^4*b - a^2*b^3)*c^2*sin(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*c^2)*sq
rt((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) - 6*(
2*(a^3*b^2 - a*b^4)*c*sin(d*x^2 + c) + 2*(a^2*b^3 - b^5)*c + ((2*a^4*b - a^2*b^3)*c^2*sin(d*x^2 + c) + (2*a^3*
b^2 - a*b^4)*c^2)*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) + 6*(2*(a^2*b^3 - b^5)*d*x^2 + 2*(a^2*b^3 - b^5)*c + 2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*
b^4)*c)*sin(d*x^2 + c) + ((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4
- (2*a^4*b - a^2*b^3)*c^2)*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) + 6*(2*(a^2*b^3 - b^5)*d*x
^2 + 2*(a^2*b^3 - b^5)*c + 2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*sin(d*x^2 + c) - ((2*a^3*b^2 - a*
b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2)*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) + 6*(2*(a^2*b^3 - b^5)*d*x^2 + 2*(a^2*b^3 - b^5)*c + 2*((a^3*b^2 -
a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*sin(d*x^2 + c) + ((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 +
((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b - a^2*b^3)*c^2)*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) + 6*(2*(a^2*b^3 - b^5)*d*x^2 + 2*(a^2*b^3 - b^5)*c + 2*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*sin(
d*x^2 + c) - ((2*a^3*b^2 - a*b^4)*d^2*x^4 - (2*a^3*b^2 - a*b^4)*c^2 + ((2*a^4*b - a^2*b^3)*d^2*x^4 - (2*a^4*b
- a^2*b^3)*c^2)*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))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3*sin
(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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**5/(a+b*csc(d*x**2+c))**2,x)

[Out]

Integral(x**5/(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^{5}}{{\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^5/(a+b*csc(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^5/(b*csc(d*x^2 + c) + a)^2, x)