Optimal. Leaf size=26 \[ b \text {Int}\left (x^2 \sec \left (c+d x^2\right ),x\right )+\frac {a x^3}{3} \]
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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 = {} \[ \int x^2 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^2+b x^2 \sec \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^3}{3}+b \int x^2 \sec \left (c+d x^2\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 0.77, size = 0, normalized size = 0.00 \[ \int x^2 \left (a+b \sec \left (c+d x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{2} \sec \left (d x^{2} + c\right ) + a x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x^{2} + c\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \sec \left (d \,x^{2}+c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a x^{3} + 2 \, b \int \frac {x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) \cos \left (d x^{2} + c\right ) + x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) \sin \left (d x^{2} + c\right ) + x^{2} \cos \left (d x^{2} + c\right )}{\cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int x^2\,\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \sec {\left (c + d x^{2} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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