\(\int \frac {x^3}{(a+b \csc (c+d x^2))^2} \, dx\) [25]

Optimal result

Integrand size = 18, antiderivative size = 616 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {b^2 \log \left (b+a \sin \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 x^2 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4290, 4276, 3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^2 \log \left (a \sin \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {b^2 x^2 \cos \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \sin \left (c+d x^2\right )+b\right )}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^4}{4 a^2} \]

[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 31

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

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 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 2747

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {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 \text {Subst}\left (\int \frac {x}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {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) \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 \text {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 ) \text {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 ) \text {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) \text {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) \text {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 \text {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 \text {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 ) \text {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 ) \text {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 \operatorname {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 \operatorname {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^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 \text {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 \text {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 \operatorname {PolyLog}\left (2,\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 \operatorname {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 \operatorname {PolyLog}\left (2,\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 \operatorname {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^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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2566\) vs. \(2(616)=1232\).

Time = 16.93 (sec) , antiderivative size = 2566, normalized size of antiderivative = 4.17 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Result too large to show} \]

[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])
] - 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])] - 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])] + 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])] - I*(2*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)*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)*PolyLog[2, -((b*(I + Tan[(c + d*x^2)/2]))/(a - I*b + Sqrt[a^2 - b^2]))] + I*(
2*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*(((2*a^2 - b
^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)/(2*(1
- I*Tan[(c + d*x^2)/2])) - ((2*a^2 - b^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)/(2*(1 - I*Tan[(c + d*x^2)/2])) - ((2*a^2 - b^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)/(2*(1 + I*Tan[(c + d*x^2)/2])) + ((2*a^2 - b^
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)/(2*(1 + I*Tan[(
c + d*x^2)/2])) + ((2*a^2 - b^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)/(2*(1 + I*Tan[(c + d*x^2)/2])) - ((2*a^2 - b^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)/(2*(1 + I*Tan[(c + d*x^2)/2])) + b*Sqrt[a^2 - b^2
]*Tan[(c + d*x^2)/2] + ((I/2)*(2*a^2 - b^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)/(I + Tan[(c + d*x^2)/2]) - ((I/2)*(2*a^2 - b^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)/(I + Tan[(c + d*x^2)/2]) + ((I/2)*b*(2*a^2 - b^2)*Log[1 - I*T
an[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]) - ((I/2)*b*(2*a^2 - b^2)
*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]) - ((I/2)*b*(
2*a^2 - b^2)*Log[1 - I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])
+ ((I/2)*b*(2*a^2 - b^2)*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(a + Sqrt[a^2 - b^2] + b*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)))))

Maple [F]

\[\int \frac {x^{3}}{{\left (a +b \csc \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

[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)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1906 vs. \(2 (526) = 1052\).

Time = 0.42 (sec) , antiderivative size = 1906, normalized size of antiderivative = 3.09 \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]

[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((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) + (-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(
(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) + (-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((-I*b*co
s(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) +
(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((-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) - ((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(-(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) + ((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(-(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) - ((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(
-(-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) + ((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(-(-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*s
in(d*x^2 + c))*sqrt((a^2 - b^2)/a^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*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*s
in(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)

Sympy [F]

\[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

int(x^3/(a + b/sin(c + d*x^2))^2,x)

[Out]

int(x^3/(a + b/sin(c + d*x^2))^2, x)