3.1.23 \(\int \frac {x^5}{(a+b \csc (c+d x^2))^2} \, dx\) [23]

Optimal. Leaf size=1124 \[ -\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,\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 {PolyLog}\left (3,\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 {PolyLog}\left (3,\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 {PolyLog}\left (3,\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 {PolyLog}\left (3,\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 )} \]

[Out]

-I*b^3*polylog(3,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3+1/6*x^6/a^2+b^2*x^2*ln(1+
a*exp(I*(d*x^2+c))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2+b^2*x^2*ln(1+a*exp(I*(d*x^2+c))/(I*b+(a^2-b^2)^(1/
2)))/a^2/(a^2-b^2)/d^2+1/2*I*b^3*x^4*ln(1-I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+I*
b*x^4*ln(1-I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/d/(-a^2+b^2)^(1/2)-2*I*b*polylog(3,I*a*exp(I*(d*x^2+
c))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2)+I*b^3*polylog(3,I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/
a^2/(-a^2+b^2)^(3/2)/d^3-b^3*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2
+b^3*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-1/2*I*b^3*x^4*ln(1-I*a*
exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-I*b^2*polylog(2,-a*exp(I*(d*x^2+c))/(I*b+(a^2-b^
2)^(1/2)))/a^2/(a^2-b^2)/d^3-1/2*b^2*x^4*cos(d*x^2+c)/a/(a^2-b^2)/d/(b+a*sin(d*x^2+c))-1/2*I*b^2*x^4/a^2/(a^2-
b^2)/d+2*I*b*polylog(3,I*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2)+2*b*x^2*polylog(2,I
*a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/(-a^2+b^2)^(1/2)-2*b*x^2*polylog(2,I*a*exp(I*(d*x^2+c))/(b+(
-a^2+b^2)^(1/2)))/a^2/d^2/(-a^2+b^2)^(1/2)-I*b*x^4*ln(1-I*a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/d/(-a^2
+b^2)^(1/2)-I*b^2*polylog(2,-a*exp(I*(d*x^2+c))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.69, antiderivative size = 1124, normalized size of antiderivative = 1.00, 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 = {4290, 4276, 3405, 3404, 2296, 2221, 2611, 2320, 6724, 4617, 2317, 2438} \begin {gather*} \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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_3\left (\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 {Li}_3\left (\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 {Li}_3\left (\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 {Li}_3\left (\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Csc[c + d*x^2])^2,x]

[Out]

((-1/2*I)*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*
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) + (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 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 2296

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 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 2320

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 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 2611

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 3404

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 3405

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 4276

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 4290

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 4617

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 6724

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]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {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) \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 \text {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 \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 9.66, size = 2033, normalized size = 1.81 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{\left (a +b \csc \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((a^4 - a^2*b^2)*d*x^6*cos(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6*cos(d*x^2 + c)^2 + (a^4 - a^2*b^2)*d
*x^6*sin(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6*sin(d*x^2 + c)^2 - 6*a*b^3*x^4*cos(d*x^2 + c) + 4*(a^3*b -
a*b^3)*d*x^6*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6 - 2*(3*a*b^3*x^4*cos(d*x^2 + c) + 2*(a^3*b - a*b^3)*d*x^6
*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6)*cos(2*d*x^2 + 2*c) - 6*((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a
^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + (a^6 - a^4*b^
2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) +
(a^6 - a^4*b^2)*d - 2*(2*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c))*integrate
(2*(2*(2*a^2*b^2 - b^4)*d*x^5*cos(d*x^2 + c)^2 + 2*(2*a^2*b^2 - b^4)*d*x^5*sin(d*x^2 + c)^2 - 2*a*b^3*x^3*cos(
d*x^2 + c) + (2*a^3*b - a*b^3)*d*x^5*sin(d*x^2 + c) - (2*a*b^3*x^3*cos(d*x^2 + c) + (2*a^3*b - a*b^3)*d*x^5*si
n(d*x^2 + c))*cos(2*d*x^2 + 2*c) + ((2*a^3*b - a*b^3)*d*x^5*cos(d*x^2 + c) - 2*a*b^3*x^3*sin(d*x^2 + c) - 2*a^
2*b^2*x^3)*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c
)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^
4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d - 2*(2*(a^5*b -
a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c)), x) + 2*(2*(a^3*b - a*b^3)*d*x^6*cos(d*x^2
+ c) - 3*a*b^3*x^4*sin(d*x^2 + c) - 3*a^2*b^2*x^4)*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^
2 + 4*(a^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + (a^6
- a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(d*x^2
+ c) + (a^6 - a^4*b^2)*d - 2*(2*(a^5*b - a^3*b^3)*d*sin(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3032 vs. \(2 (966) = 1932\).
time = 3.00, size = 3032, normalized size = 2.70 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/12*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6*sin(d*x^2 + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d^3*x^6 - 6*(a^3*b^2 -
a*b^4)*d^2*x^4*cos(d*x^2 + c) + 6*(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, -(I*b*cos(d*x^2 + c) + b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^
2)/a^2))/a) - 6*(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, -(I*
b*cos(d*x^2 + c) + b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) + 6*(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, -(-I*b*cos(d*x^2 + c)
+ b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) - 6*(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, -(-I*b*cos(d*x^2 + c) + b*sin(d*x^2 + c
) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) - 6*(I*a^2*b^3 - I*b^5 + (I*a^3*b^2 - I*
a*b^4)*sin(d*x^2 + c) + (-I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) - I*(2*a^3*b^2 - a*b^4)*d*x^2)*sqrt((a^2
- b^2)/a^2))*dilog((I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2
- b^2)/a^2) - a)/a + 1) - 6*(I*a^2*b^3 - I*b^5 + (I*a^3*b^2 - I*a*b^4)*sin(d*x^2 + c) + (I*(2*a^4*b - a^2*b^3)
*d*x^2*sin(d*x^2 + c) + I*(2*a^3*b^2 - a*b^4)*d*x^2)*sqrt((a^2 - b^2)/a^2))*dilog((I*b*cos(d*x^2 + c) - b*sin(
d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) - 6*(-I*a^2*b^3 + I*b^5
+ (-I*a^3*b^2 + I*a*b^4)*sin(d*x^2 + c) + (I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) + I*(2*a^3*b^2 - a*b^4)
*d*x^2)*sqrt((a^2 - b^2)/a^2))*dilog((-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x
^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) - 6*(-I*a^2*b^3 + I*b^5 + (-I*a^3*b^2 + I*a*b^4)*sin(d*x^2 + c) + (
-I*(2*a^4*b - a^2*b^3)*d*x^2*sin(d*x^2 + c) - I*(2*a^3*b^2 - a*b^4)*d*x^2)*sqrt((a^2 - b^2)/a^2))*dilog((-I*b*
cos(d*x^2 + c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1)
- 3*(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) - 3*(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) - 3*(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) - 3*(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) + 3*(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))*sqr
t((a^2 - b^2)/a^2))*log(-(I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt
((a^2 - b^2)/a^2) - a)/a) + 3*(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(-(I*b*cos(d*x^2 + c) - b*sin
(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a) + 3*(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(-(-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c
))*sqrt((a^2 - b^2)/a^2) - a)/a) + 3*(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(-(-I*b*cos(d*x^2 + c)
- b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b/sin(c + d*x^2))^2,x)

[Out]

int(x^5/(a + b/sin(c + d*x^2))^2, x)

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