Optimal. Leaf size=52 \[ \frac {\sqrt {1+a^2 x^2}}{3 a^3}-\frac {\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5776, 272, 45}
\begin {gather*} -\frac {\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac {\sqrt {a^2 x^2+1}}{3 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 5776
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+a^2 x^2}}{3 a^3}-\frac {\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac {1}{3} x^3 \sinh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.79 \begin {gather*} \frac {1}{9} \left (\frac {\left (2-a^2 x^2\right ) \sqrt {1+a^2 x^2}}{a^3}+3 x^3 \sinh ^{-1}(a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 50, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{9}+\frac {2 \sqrt {a^{2} x^{2}+1}}{9}}{a^{3}}\) | \(50\) |
default | \(\frac {\frac {a^{3} x^{3} \arcsinh \left (a x \right )}{3}-\frac {a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{9}+\frac {2 \sqrt {a^{2} x^{2}+1}}{9}}{a^{3}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 48, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 52, normalized size = 1.00 \begin {gather*} \frac {3 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )}}{9 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 48, normalized size = 0.92 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {asinh}{\left (a x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{9 a} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^2\,\mathrm {asinh}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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