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

Optimal result

Integrand size = 18, antiderivative size = 596 \[ \int \frac {x^3}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {b^2 \log \left (b+a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^2 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )} \]

[Out]

1/4*x^4/a^2+1/2*b^2*ln(b+a*cos(d*x^2+c))/a^2/(a^2-b^2)/d^2-1/2*I*b^3*x^2*ln(1+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+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,-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,-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*sin(d*x^2+c)/a/(a^2-b^2)/d/(b+a*c
os(d*x^2+c))+I*b*x^2*ln(1+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+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,-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,-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 = 1.34 (sec) , antiderivative size = 596, 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 = {4289, 4276, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {x^3}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^2 \log \left (a \cos \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}+\frac {b^2 x^2 \sin \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cos \left (c+d x^2\right )+b\right )}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^4}{4 a^2} \]

[In]

Int[x^3/(a + b*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 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 3402

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 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 4289

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]

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

Time = 11.66 (sec) , antiderivative size = 1118, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\frac {\left (-c+d x^2\right ) \left (c+d x^2\right ) \left (b+a \cos \left (c+d x^2\right )\right )^2 \sec ^2\left (c+d x^2\right )}{4 a^2 d^2 \left (a+b \sec \left (c+d x^2\right )\right )^2}+\frac {\left (b+a \cos \left (c+d x^2\right )\right ) \sec ^2\left (c+d x^2\right ) \left (b^2 c \sin \left (c+d x^2\right )-b^2 \left (c+d x^2\right ) \sin \left (c+d x^2\right )\right )}{2 a (-a+b) (a+b) d^2 \left (a+b \sec \left (c+d x^2\right )\right )^2}+\frac {b \cos ^2\left (\frac {1}{2} \left (c+d x^2\right )\right ) \left (b+a \cos \left (c+d x^2\right )\right ) \left (2 \left (2 a^2-b^2\right ) c \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )-\sqrt {a-b} b \sqrt {a+b} \log \left (\sec ^2\left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+\sqrt {a-b} b \sqrt {a+b} \log \left (\left (b+a \cos \left (c+d x^2\right )\right ) \sec ^2\left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+i \left (2 a^2-b^2\right ) \left (\log \left (1-i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right ) \log \left (\frac {\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{i \sqrt {a-b}+\sqrt {a+b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )\right )-i \left (2 a^2-b^2\right ) \left (\log \left (1-i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right ) \log \left (\frac {i \left (\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )\right )+i \left (2 a^2-b^2\right ) \left (\log \left (1+i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right ) \log \left (\frac {\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{i \sqrt {a-b}+\sqrt {a+b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1+i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )\right )-i \left (2 a^2-b^2\right ) \left (\log \left (1+i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right ) \log \left (\frac {i \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1+i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )\right )\right ) \sec ^2\left (c+d x^2\right ) \left (\left (2 a^2-b^2\right ) d x^2+a b \sin \left (c+d x^2\right )\right ) \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right ) \left (\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{2 a^2 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right ) d^2 \left (a+b \sec \left (c+d x^2\right )\right )^2 \left (-\left (\left (2 a^2-b^2\right ) \left (c-i \log \left (1-i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+i \log \left (1+i \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )\right )\right )+a b \sin \left (c+d x^2\right )\right )} \]

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

Maple [F]

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

[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)

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. 1928 vs. \(2 (522) = 1044\).

Time = 0.44 (sec) , antiderivative size = 1928, normalized size of antiderivative = 3.23 \[ \int \frac {x^3}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]

[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
(-(b*cos(d*x^2 + c) + I*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 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-(b*cos(d*x^2
+ c) + I*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 + (2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-(b*cos(d*x^2 + c) - I*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 +
(2*a^4*b - a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*dilog(-(b*cos(d*x^2 + c) - I*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) + (-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((b*cos(d*x^2 + c) + I*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) + (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((b*cos(d*x^2 + c) + I*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) + (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)*l
og((b*cos(d*x^2 + c) - I*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) + (-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((b*cos(d*x^2 + c) - I*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^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*co
s(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)*c
os(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))*lo
g(-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*cos(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2)

Sympy [F]

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

[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)

Maxima [F(-2)]

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

[In]

integrate(x^3/(a+b*sec(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*b^2-4*a^2>0)', see `assume?`
for more de

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

int(x^3/(a + b/cos(c + d*x^2))^2,x)

[Out]

int(x^3/(a + b/cos(c + d*x^2))^2, x)