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

Optimal. Leaf size=596 \[ \frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^2 \log \left (a \cos \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{i b^3 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{i b^3 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^2 \sin \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cos \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.20394, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4204, 4191, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac{b \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^2 \log \left (a \cos \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d \sqrt{b^2-a^2}}-\frac{i b^3 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac{i b x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d \sqrt{b^2-a^2}}+\frac{i b^3 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^2 \sin \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cos \left (c+d x^2\right )+b\right )}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sec[c + d*x^2])^2,x]

[Out]

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

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[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 3321

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

Rule 2264

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

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2668

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

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 9.13948, size = 1069, normalized size = 1.79 \[ \frac{\left (b+a \cos \left (d x^2+c\right )\right ) \left (b^2 c \sin \left (d x^2+c\right )-b^2 \left (d x^2+c\right ) \sin \left (d x^2+c\right )\right ) \sec ^2\left (d x^2+c\right )}{2 a (b-a) (a+b) d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2}+\frac{b \cos ^2\left (\frac{1}{2} \left (d x^2+c\right )\right ) \left (b+a \cos \left (d x^2+c\right )\right ) \left (-\frac{2 \left (2 a^2-b^2\right ) c \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )}{\sqrt{-a-b}}\right )}{\sqrt{-a-b} \sqrt{a-b}}+b \left (\log \left (-\left (b+a \cos \left (d x^2+c\right )\right ) \sec ^2\left (\frac{1}{2} \left (d x^2+c\right )\right )\right )-\log \left (\sec ^2\left (\frac{1}{2} \left (d x^2+c\right )\right )\right )\right )-\frac{i \left (2 a^2-b^2\right ) \left (\log \left (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right ) \log \left (\frac{i \left (\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\log \left (1-i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right ) \log \left (\frac{\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )}{i \sqrt{a-b}+\sqrt{a+b}}\right )+\log \left (1-i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right ) \log \left (\frac{i \left (\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+\sqrt{a+b}\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\log \left (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right ) \log \left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+\sqrt{a+b}}{i \sqrt{a-b}+\sqrt{a+b}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (1-i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt{a-b}-i \sqrt{a+b}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (1-i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right )}{\sqrt{a-b}-i \sqrt{a+b}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )\right )}{\sqrt{a-b} \sqrt{a+b}}\right ) \left (\left (2 a^2-b^2\right ) d x^2+a b \sin \left (d x^2+c\right )\right ) \left (\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right ) \left (\sqrt{a-b} \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+\sqrt{a+b}\right ) \sec ^2\left (d x^2+c\right )}{2 a^2 \left (a^2-b^2\right ) d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2 \left (a b \sin \left (d x^2+c\right )-\left (2 a^2-b^2\right ) \left (c-i \log \left (1-i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )\right )+i \log \left (i \tan \left (\frac{1}{2} \left (d x^2+c\right )\right )+1\right )\right )\right )}+\frac{\left (d x^2-c\right ) \left (d x^2+c\right ) \left (b+a \cos \left (d x^2+c\right )\right )^2 \sec ^2\left (d x^2+c\right )}{4 a^2 d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( a+b\sec \left ( d{x}^{2}+c \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.13181, size = 4228, normalized size = 7.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*sec(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*cos(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*sin(d*x^2 + c) - (2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog
(-1/2*(2*b*cos(d*x^2 + c) + 2*I*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)
/a^2) + 2*a)/a + 1) + (2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-1
/2*(2*b*cos(d*x^2 + c) + 2*I*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^
2) + 2*a)/a + 1) - (2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-1/2*
(2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)
+ 2*a)/a + 1) + (2*a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-1/2*(2*
b*cos(d*x^2 + c) - 2*I*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) + (-I*(2*a^3*b^2 - a*b^4)*d*x^2 - I*(2*a^3*b^2 - a*b^4)*c + (-I*(2*a^4*b - a^2*b^3)*d*x^2 - I*(2*a^
4*b - a^2*b^3)*c)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*log(1/2*(2*b*cos(d*x^2 + c) + 2*I*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) + (I*(2*a^3*b^2 - a*b^4)*d*x^2 + I*
(2*a^3*b^2 - a*b^4)*c + (I*(2*a^4*b - a^2*b^3)*d*x^2 + I*(2*a^4*b - a^2*b^3)*c)*cos(d*x^2 + c))*sqrt(-(a^2 - b
^2)/a^2)*log(1/2*(2*b*cos(d*x^2 + c) + 2*I*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) + (I*(2*a^3*b^2 - a*b^4)*d*x^2 + I*(2*a^3*b^2 - a*b^4)*c + (I*(2*a^4*b - a^2*b^3)*d
*x^2 + I*(2*a^4*b - a^2*b^3)*c)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*log(1/2*(2*b*cos(d*x^2 + c) - 2*I*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) + (-I*(2*a^3*b^2 - a*
b^4)*d*x^2 - I*(2*a^3*b^2 - a*b^4)*c + (-I*(2*a^4*b - a^2*b^3)*d*x^2 - I*(2*a^4*b - a^2*b^3)*c)*cos(d*x^2 + c)
)*sqrt(-(a^2 - b^2)/a^2)*log(1/2*(2*b*cos(d*x^2 + c) - 2*I*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^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cos(d*x^2 + c) + (-I*(2*a^4*b
- a^2*b^3)*c*cos(d*x^2 + c) - I*(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*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cos(d*x^2 + c) + (I*(2
*a^4*b - a^2*b^3)*c*cos(d*x^2 + c) + I*(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*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cos(d*x^2 + c)
+ (-I*(2*a^4*b - a^2*b^3)*c*cos(d*x^2 + c) - I*(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*b) + (a^2*b^3 - b^5 + (a^3*b^2 - a*b^4)*cos(d*
x^2 + c) + (I*(2*a^4*b - a^2*b^3)*c*cos(d*x^2 + c) + I*(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*b))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*c
os(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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