3.1.19 \(\int \frac {x^3}{(a+b \tan (c+d x^2))^2} \, dx\) [19]

Optimal. Leaf size=202 \[ -\frac {x^4}{4 \left (a^2+b^2\right )}+\frac {\left (b+2 a d x^2\right )^2}{8 a (a+i b) \left (a^2+b^2\right ) d^2}+\frac {b \left (b+2 a d x^2\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {i a b \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )} \]

[Out]

-1/4*x^4/(a^2+b^2)+1/8*(2*a*d*x^2+b)^2/a/(a+I*b)/(a^2+b^2)/d^2+1/2*b*(2*a*d*x^2+b)*ln(1+(a^2+b^2)*exp(2*I*(d*x
^2+c))/(a+I*b)^2)/(a^2+b^2)^2/d^2-1/2*I*a*b*polylog(2,-(a^2+b^2)*exp(2*I*(d*x^2+c))/(a+I*b)^2)/(a^2+b^2)^2/d^2
-1/2*b*x^2/(a^2+b^2)/d/(a+b*tan(d*x^2+c))

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Rubi [A]
time = 0.23, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3832, 3814, 3813, 2221, 2317, 2438} \begin {gather*} -\frac {i a b \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (d x^2+c\right )}}{(a+i b)^2}\right )}{2 d^2 \left (a^2+b^2\right )^2}+\frac {b \left (2 a d x^2+b\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 d^2 \left (a^2+b^2\right )^2}-\frac {b x^2}{2 d \left (a^2+b^2\right ) \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {\left (2 a d x^2+b\right )^2}{8 a d^2 (a+i b) \left (a^2+b^2\right )}-\frac {x^4}{4 \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Tan[c + d*x^2])^2,x]

[Out]

-1/4*x^4/(a^2 + b^2) + (b + 2*a*d*x^2)^2/(8*a*(a + I*b)*(a^2 + b^2)*d^2) + (b*(b + 2*a*d*x^2)*Log[1 + ((a^2 +
b^2)*E^((2*I)*(c + d*x^2)))/(a + I*b)^2])/(2*(a^2 + b^2)^2*d^2) - ((I/2)*a*b*PolyLog[2, -(((a^2 + b^2)*E^((2*I
)*(c + d*x^2)))/(a + I*b)^2)])/((a^2 + b^2)^2*d^2) - (b*x^2)/(2*(a^2 + b^2)*d*(a + b*Tan[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 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 3813

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

Rule 3814

Int[((c_.) + (d_.)*(x_))/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[-(c + d*x)^2/(2*d*(a^2 +
b^2)), x] + (Dist[1/(f*(a^2 + b^2)), Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x], x] - Simp[b*((c
+ d*x)/(f*(a^2 + b^2)*(a + b*Tan[e + f*x]))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]

Rule 3832

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[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*} \int \frac {x^3}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b \tan (c+d x))^2} \, dx,x,x^2\right )\\ &=-\frac {x^4}{4 \left (a^2+b^2\right )}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {\text {Subst}\left (\int \frac {b+2 a d x}{a+b \tan (c+d x)} \, dx,x,x^2\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac {x^4}{4 \left (a^2+b^2\right )}+\frac {\left (b+2 a d x^2\right )^2}{8 a (a+i b) \left (a^2+b^2\right ) d^2}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {(i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} (b+2 a d x)}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,x^2\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac {x^4}{4 \left (a^2+b^2\right )}+\frac {\left (b+2 a d x^2\right )^2}{8 a (a+i b) \left (a^2+b^2\right ) d^2}+\frac {b \left (b+2 a d x^2\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}-\frac {(a b) \text {Subst}\left (\int \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,x^2\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {x^4}{4 \left (a^2+b^2\right )}+\frac {\left (b+2 a d x^2\right )^2}{8 a (a+i b) \left (a^2+b^2\right ) d^2}+\frac {b \left (b+2 a d x^2\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}+\frac {(i a b) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d x^2\right )}\right )}{2 \left (a^2+b^2\right )^2 d^2}\\ &=-\frac {x^4}{4 \left (a^2+b^2\right )}+\frac {\left (b+2 a d x^2\right )^2}{8 a (a+i b) \left (a^2+b^2\right ) d^2}+\frac {b \left (b+2 a d x^2\right ) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {i a b \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d x^2\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {b x^2}{2 \left (a^2+b^2\right ) d \left (a+b \tan \left (c+d x^2\right )\right )}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(703\) vs. \(2(202)=404\).
time = 6.76, size = 703, normalized size = 3.48 \begin {gather*} \frac {\left (-c+d x^2\right ) \left (c+d x^2\right ) \sec ^2\left (c+d x^2\right ) \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )^2}{4 (a-i b) (a+i b) d^2 \left (a+b \tan \left (c+d x^2\right )\right )^2}+\frac {b^2 \left (-b \left (c+d x^2\right )+a \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )\right ) \sec ^2\left (c+d x^2\right ) \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )^2}{2 a (a-i b) (a+i b) \left (a^2+b^2\right ) d^2 \left (a+b \tan \left (c+d x^2\right )\right )^2}-\frac {b c \left (-b \left (c+d x^2\right )+a \log \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )\right ) \sec ^2\left (c+d x^2\right ) \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )^2}{(a-i b) (a+i b) \left (a^2+b^2\right ) d^2 \left (a+b \tan \left (c+d x^2\right )\right )^2}-\frac {\left (e^{i \text {ArcTan}\left (\frac {a}{b}\right )} \left (c+d x^2\right )^2+\frac {a \left (i \left (c+d x^2\right ) \left (-\pi +2 \text {ArcTan}\left (\frac {a}{b}\right )\right )-\pi \log \left (1+e^{-2 i \left (c+d x^2\right )}\right )-2 \left (c+d x^2+\text {ArcTan}\left (\frac {a}{b}\right )\right ) \log \left (1-e^{2 i \left (c+d x^2+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )+\pi \log \left (\cos \left (c+d x^2\right )\right )+2 \text {ArcTan}\left (\frac {a}{b}\right ) \log \left (\sin \left (c+d x^2+\text {ArcTan}\left (\frac {a}{b}\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (c+d x^2+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )\right )}{\sqrt {1+\frac {a^2}{b^2}} b}\right ) \sec ^2\left (c+d x^2\right ) \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\right )^2}{2 (a-i b) (a+i b) \sqrt {\frac {a^2+b^2}{b^2}} d^2 \left (a+b \tan \left (c+d x^2\right )\right )^2}+\frac {\sec ^2\left (c+d x^2\right ) \left (a \cos \left (c+d x^2\right )+b \sin \left (c+d x^2\right )\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-i b) (a+i b) d^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (179) = 358\).
time = 0.40, size = 1001, normalized size = 4.96 \begin {gather*} \frac {{\left (a^{3} - i \, a^{2} b + a b^{2} - i \, b^{3}\right )} d^{2} x^{4} - 2 \, {\left (-i \, a b^{2} + b^{3} + {\left (-i \, a b^{2} - b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a b^{2} - i \, b^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \arctan \left (-b \cos \left (2 \, d x^{2} + 2 \, c\right ) + a \sin \left (2 \, d x^{2} + 2 \, c\right ) + b, a \cos \left (2 \, d x^{2} + 2 \, c\right ) + b \sin \left (2 \, d x^{2} + 2 \, c\right ) + a\right ) - 4 \, {\left ({\left (i \, a^{2} b + a b^{2}\right )} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (a^{2} b - i \, a b^{2}\right )} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (i \, a^{2} b - a b^{2}\right )} d x^{2}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (a^{3} - 3 i \, a^{2} b - 3 \, a b^{2} + i \, b^{3}\right )} d^{2} x^{4} - 4 \, {\left (i \, a b^{2} + b^{3}\right )} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) - 2 \, {\left (i \, a^{2} b - a b^{2} + {\left (i \, a^{2} b + a b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (a^{2} b - i \, a b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d x^{2} + 2 i \, c\right )}}{-i \, a + b}\right ) + {\left (a b^{2} + i \, b^{3} + {\left (a b^{2} - i \, b^{3}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (i \, a b^{2} + b^{3}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )\right ) + 2 \, {\left ({\left (a^{2} b - i \, a b^{2}\right )} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (-i \, a^{2} b - a b^{2}\right )} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} b + i \, a b^{2}\right )} d x^{2}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (i \, a^{3} + 3 \, a^{2} b - 3 i \, a b^{2} - b^{3}\right )} d^{2} x^{4} + 4 \, {\left (a b^{2} - i \, b^{3}\right )} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{4 \, {\left ({\left (a^{5} - i \, a^{4} b + 2 \, a^{3} b^{2} - 2 i \, a^{2} b^{3} + a b^{4} - i \, b^{5}\right )} d^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (-i \, a^{5} - a^{4} b - 2 i \, a^{3} b^{2} - 2 \, a^{2} b^{3} - i \, a b^{4} - b^{5}\right )} d^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a^{5} + i \, a^{4} b + 2 \, a^{3} b^{2} + 2 i \, a^{2} b^{3} + a b^{4} + i \, b^{5}\right )} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^2*x^4 - 2*(-I*a*b^2 + b^3 + (-I*a*b^2 - b^3)*cos(2*d*x^2 + 2*c) + (a*b^
2 - I*b^3)*sin(2*d*x^2 + 2*c))*arctan2(-b*cos(2*d*x^2 + 2*c) + a*sin(2*d*x^2 + 2*c) + b, a*cos(2*d*x^2 + 2*c)
+ b*sin(2*d*x^2 + 2*c) + a) - 4*((I*a^2*b + a*b^2)*d*x^2*cos(2*d*x^2 + 2*c) - (a^2*b - I*a*b^2)*d*x^2*sin(2*d*
x^2 + 2*c) + (I*a^2*b - a*b^2)*d*x^2)*arctan2((2*a*b*cos(2*d*x^2 + 2*c) - (a^2 - b^2)*sin(2*d*x^2 + 2*c))/(a^2
+ b^2), (2*a*b*sin(2*d*x^2 + 2*c) + a^2 + b^2 + (a^2 - b^2)*cos(2*d*x^2 + 2*c))/(a^2 + b^2)) + ((a^3 - 3*I*a^
2*b - 3*a*b^2 + I*b^3)*d^2*x^4 - 4*(I*a*b^2 + b^3)*d*x^2)*cos(2*d*x^2 + 2*c) - 2*(I*a^2*b - a*b^2 + (I*a^2*b +
a*b^2)*cos(2*d*x^2 + 2*c) - (a^2*b - I*a*b^2)*sin(2*d*x^2 + 2*c))*dilog((I*a + b)*e^(2*I*d*x^2 + 2*I*c)/(-I*a
+ b)) + (a*b^2 + I*b^3 + (a*b^2 - I*b^3)*cos(2*d*x^2 + 2*c) + (I*a*b^2 + b^3)*sin(2*d*x^2 + 2*c))*log((a^2 +
b^2)*cos(2*d*x^2 + 2*c)^2 + 4*a*b*sin(2*d*x^2 + 2*c) + (a^2 + b^2)*sin(2*d*x^2 + 2*c)^2 + a^2 + b^2 + 2*(a^2 -
b^2)*cos(2*d*x^2 + 2*c)) + 2*((a^2*b - I*a*b^2)*d*x^2*cos(2*d*x^2 + 2*c) - (-I*a^2*b - a*b^2)*d*x^2*sin(2*d*x
^2 + 2*c) + (a^2*b + I*a*b^2)*d*x^2)*log(((a^2 + b^2)*cos(2*d*x^2 + 2*c)^2 + 4*a*b*sin(2*d*x^2 + 2*c) + (a^2 +
b^2)*sin(2*d*x^2 + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^2 + 2*c))/(a^2 + b^2)) + ((I*a^3 + 3*a^2*b -
3*I*a*b^2 - b^3)*d^2*x^4 + 4*(a*b^2 - I*b^3)*d*x^2)*sin(2*d*x^2 + 2*c))/((a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*
b^3 + a*b^4 - I*b^5)*d^2*cos(2*d*x^2 + 2*c) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*d^2*s
in(2*d*x^2 + 2*c) + (a^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b^5)*d^2)

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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. 800 vs. \(2 (179) = 358\).
time = 0.43, size = 800, normalized size = 3.96 \begin {gather*} \frac {{\left (a^{3} - a b^{2}\right )} d^{2} x^{4} - 2 \, b^{3} d x^{2} + {\left (i \, a b^{2} \tan \left (d x^{2} + c\right ) + i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{2} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) + {\left (-i \, a b^{2} \tan \left (d x^{2} + c\right ) - i \, a^{2} b\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{2} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} + a^{2} + b^{2}} + 1\right ) + 2 \, {\left (a^{2} b d x^{2} + a^{2} b c + {\left (a b^{2} d x^{2} + a b^{2} c\right )} \tan \left (d x^{2} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (d x^{2} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} + a^{2} + b^{2}}\right ) + 2 \, {\left (a^{2} b d x^{2} + a^{2} b c + {\left (a b^{2} d x^{2} + a b^{2} c\right )} \tan \left (d x^{2} + c\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (d x^{2} + c\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} + a^{2} + b^{2}}\right ) - {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d x^{2} + c\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{2} + c\right )}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) - {\left (2 \, a^{2} b c - a b^{2} + {\left (2 \, a b^{2} c - b^{3}\right )} \tan \left (d x^{2} + c\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (d x^{2} + c\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (d x^{2} + c\right )}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) + {\left ({\left (a^{2} b - b^{3}\right )} d^{2} x^{4} + 2 \, a b^{2} d x^{2}\right )} \tan \left (d x^{2} + c\right )}{4 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{2} \tan \left (d x^{2} + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^3 - a*b^2)*d^2*x^4 - 2*b^3*d*x^2 + (I*a*b^2*tan(d*x^2 + c) + I*a^2*b)*dilog(2*((I*a*b - b^2)*tan(d*x^2
+ c)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(d*x^2 + c))/((a^2 + b^2)*tan(d*x^2 + c)^2 + a^2 + b^2) + 1
) + (-I*a*b^2*tan(d*x^2 + c) - I*a^2*b)*dilog(2*((-I*a*b - b^2)*tan(d*x^2 + c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a
*b + I*b^2)*tan(d*x^2 + c))/((a^2 + b^2)*tan(d*x^2 + c)^2 + a^2 + b^2) + 1) + 2*(a^2*b*d*x^2 + a^2*b*c + (a*b^
2*d*x^2 + a*b^2*c)*tan(d*x^2 + c))*log(-2*((I*a*b - b^2)*tan(d*x^2 + c)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b
^2)*tan(d*x^2 + c))/((a^2 + b^2)*tan(d*x^2 + c)^2 + a^2 + b^2)) + 2*(a^2*b*d*x^2 + a^2*b*c + (a*b^2*d*x^2 + a*
b^2*c)*tan(d*x^2 + c))*log(-2*((-I*a*b - b^2)*tan(d*x^2 + c)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(d*
x^2 + c))/((a^2 + b^2)*tan(d*x^2 + c)^2 + a^2 + b^2)) - (2*a^2*b*c - a*b^2 + (2*a*b^2*c - b^3)*tan(d*x^2 + c))
*log(((I*a*b + b^2)*tan(d*x^2 + c)^2 - a^2 + I*a*b + (I*a^2 + I*b^2)*tan(d*x^2 + c))/(tan(d*x^2 + c)^2 + 1)) -
(2*a^2*b*c - a*b^2 + (2*a*b^2*c - b^3)*tan(d*x^2 + c))*log(((I*a*b - b^2)*tan(d*x^2 + c)^2 + a^2 + I*a*b + (I
*a^2 + I*b^2)*tan(d*x^2 + c))/(tan(d*x^2 + c)^2 + 1)) + ((a^2*b - b^3)*d^2*x^4 + 2*a*b^2*d*x^2)*tan(d*x^2 + c)
)/((a^4*b + 2*a^2*b^3 + b^5)*d^2*tan(d*x^2 + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^3/(b*tan(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^3}{{\left (a+b\,\mathrm {tan}\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^3/(a + b*tan(c + d*x^2))^2,x)

[Out]

int(x^3/(a + b*tan(c + d*x^2))^2, x)

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