Optimal. Leaf size=56 \[ \frac {1}{6} a \sqrt {1+\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1+\frac {a^2}{x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5870, 6419,
272, 44, 65, 214} \begin {gather*} \frac {1}{6} a x^2 \sqrt {\frac {a^2}{x^2}+1}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {\frac {a^2}{x^2}+1}\right )+\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 5870
Rule 6419
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x^2 \text {csch}^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )+\frac {1}{3} a \int \frac {x}{\sqrt {1+\frac {a^2}{x^2}}} \, dx\\ &=\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{6} a \sqrt {1+\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )+\frac {1}{12} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{6} a \sqrt {1+\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )+\frac {1}{6} a \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+\frac {a^2}{x^2}}\right )\\ &=\frac {1}{6} a \sqrt {1+\frac {a^2}{x^2}} x^2+\frac {1}{3} x^3 \text {csch}^{-1}\left (\frac {x}{a}\right )-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1+\frac {a^2}{x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 1.02 \begin {gather*} \frac {1}{6} \left (a \sqrt {1+\frac {a^2}{x^2}} x^2+2 x^3 \sinh ^{-1}\left (\frac {a}{x}\right )-a^3 \log \left (\left (1+\sqrt {1+\frac {a^2}{x^2}}\right ) x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 54, normalized size = 0.96
method | result | size |
derivativedivides | \(-a^{3} \left (-\frac {x^{3} \arcsinh \left (\frac {a}{x}\right )}{3 a^{3}}-\frac {x^{2} \sqrt {1+\frac {a^{2}}{x^{2}}}}{6 a^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1+\frac {a^{2}}{x^{2}}}}\right )}{6}\right )\) | \(54\) |
default | \(-a^{3} \left (-\frac {x^{3} \arcsinh \left (\frac {a}{x}\right )}{3 a^{3}}-\frac {x^{2} \sqrt {1+\frac {a^{2}}{x^{2}}}}{6 a^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1+\frac {a^{2}}{x^{2}}}}\right )}{6}\right )\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 69, normalized size = 1.23 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arsinh}\left (\frac {a}{x}\right ) - \frac {1}{12} \, {\left (a^{2} \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + 1\right ) - a^{2} \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} - 1\right ) - 2 \, x^{2} \sqrt {\frac {a^{2}}{x^{2}} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs.
\(2 (46) = 92\).
time = 0.38, size = 122, normalized size = 2.18 \begin {gather*} \frac {1}{6} \, a^{3} \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - x\right ) + \frac {1}{6} \, a x^{2} \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + \frac {1}{3} \, {\left (x^{3} - 1\right )} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \frac {1}{3} \, \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a - x\right ) - \frac {1}{3} \, \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - a - x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {asinh}{\left (\frac {a}{x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 74, normalized size = 1.32 \begin {gather*} -\frac {1}{6} \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {1}{3} \, x^{3} \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right ) + \frac {a^{3} \log \left (-x + \sqrt {a^{2} + x^{2}}\right )}{6 \, \mathrm {sgn}\left (x\right )} + \frac {\sqrt {a^{2} + x^{2}} a x}{6 \, \mathrm {sgn}\left (x\right )} \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 (\frac {a}{x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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