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

Optimal. Leaf size=26 \[ b \text {Int}\left (x^2 \tan \left (c+d x^2\right ),x\right )+\frac {a x^3}{3} \]

[Out]

1/3*a*x^3+b*Unintegrable(x^2*tan(d*x^2+c),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 = {} \[ \int x^2 \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*Tan[c + d*x^2]),x]

[Out]

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

Rubi steps

\begin {align*} \int x^2 \left (a+b \tan \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^2+b x^2 \tan \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^3}{3}+b \int x^2 \tan \left (c+d x^2\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 2.03, size = 0, normalized size = 0.00 \[ \int x^2 \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{2} \tan \left (d x^{2} + c\right ) + a x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.45, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \tan \left (d \,x^{2}+c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int x^2\,\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*tan(c + d*x^2)),x)

[Out]

int(x^2*(a + b*tan(c + d*x^2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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