Optimal. Leaf size=31 \[ \frac {1}{3} x^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6336, 30, 264} \[ \frac {1}{3} x^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 264
Rule 6336
Rubi steps
\begin {align*} \int e^{\text {csch}^{-1}(a x)} x^2 \, dx &=\frac {\int x \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^2}} x^2 \, dx\\ &=\frac {x^2}{2 a}+\frac {1}{3} \left (1+\frac {1}{a^2 x^2}\right )^{3/2} x^3\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 1.23 \[ \frac {2 \sqrt {\frac {1}{a^2 x^2}+1} \left (a^2 x^3+x\right )+3 a x^2}{6 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 41, normalized size = 1.32 \[ \frac {3 \, a x^{2} + 2 \, {\left (a^{2} x^{3} + x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}}}{6 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 44, normalized size = 1.42 \[ \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{3 \, a^{4}} + \frac {a^{2} x^{2} + 1}{2 \, a^{3}} - \frac {{\left | a \right |} \mathrm {sgn}\relax (x)}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 1.39 \[ \frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (a^{2} x^{2}+1\right )}{3 a^{2}}+\frac {x^{2}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 25, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {x^{2}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 33, normalized size = 1.06 \[ \left (\frac {x}{3\,a^2}+\frac {x^3}{3}\right )\,\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {x^2}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.71, size = 41, normalized size = 1.32 \[ \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{3 a} + \frac {x^{2}}{2 a} + \frac {\sqrt {a^{2} x^{2} + 1}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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