3.8 \(\int x^2 (a+b \tan (c+d x^2))^2 \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

Unintegrable(x^2*(a+b*tan(d*x^2+c))^2,x)

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*Tan[c + d*x^2])^2,x]

[Out]

Defer[Int][x^2*(a + b*Tan[c + d*x^2])^2, x]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx &=\int x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx\\ \end {align*}

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Mathematica [A]  time = 2.62, size = 0, normalized size = 0.00 \[ \int x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b x^{2} \tan \left (d x^{2} + c\right ) + a^{2} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x^2*tan(d*x^2 + c)^2 + 2*a*b*x^2*tan(d*x^2 + c) + a^2*x^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} x^{3} - \frac {b^{2} d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + b^{2} d x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, b^{2} d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + b^{2} d x^{3} - 3 \, b^{2} x \sin \left (2 \, d x^{2} + 2 \, c\right ) - \frac {3 \, {\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} {\left (4 \, a d \int \frac {x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{\cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1}\,{d x} - b \int \frac {\sin \left (2 \, d x^{2} + 2 \, c\right )}{\cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1}\,{d x}\right )} b}{d}}{3 \, {\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int x^2\,{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*tan(c + d*x^2))^2,x)

[Out]

int(x^2*(a + b*tan(c + d*x^2))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*tan(c + d*x**2))**2, x)

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