Optimal. Leaf size=50 \[ \frac {\sinh ^{-1}\left (a x^2\right )}{8 a^2}-\frac {x^2 \sqrt {a^2 x^4+1}}{8 a}+\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5902, 12, 275, 321, 215} \[ -\frac {x^2 \sqrt {a^2 x^4+1}}{8 a}+\frac {\sinh ^{-1}\left (a x^2\right )}{8 a^2}+\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 275
Rule 321
Rule 5902
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right )-\frac {1}{4} \int \frac {2 a x^5}{\sqrt {1+a^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right )-\frac {1}{2} a \int \frac {x^5}{\sqrt {1+a^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right )-\frac {1}{4} a \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \sqrt {1+a^2 x^4}}{8 a}+\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx,x,x^2\right )}{8 a}\\ &=-\frac {x^2 \sqrt {1+a^2 x^4}}{8 a}+\frac {\sinh ^{-1}\left (a x^2\right )}{8 a^2}+\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.88 \[ \frac {\left (2 a^2 x^4+1\right ) \sinh ^{-1}\left (a x^2\right )-a x^2 \sqrt {a^2 x^4+1}}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 52, normalized size = 1.04 \[ -\frac {\sqrt {a^{2} x^{4} + 1} a x^{2} - {\left (2 \, a^{2} x^{4} + 1\right )} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 74, normalized size = 1.48 \[ \frac {1}{4} \, x^{4} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \frac {1}{8} \, a {\left (\frac {\sqrt {a^{2} x^{4} + 1} x^{2}}{a^{2}} + \frac {\log \left (-x^{2} {\left | a \right |} + \sqrt {a^{2} x^{4} + 1}\right )}{a^{2} {\left | a \right |}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 67, normalized size = 1.34 \[ \frac {x^{4} \arcsinh \left (a \,x^{2}\right )}{4}-\frac {x^{2} \sqrt {a^{2} x^{4}+1}}{8 a}+\frac {\ln \left (\frac {a^{2} x^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{4}+1}\right )}{8 a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 102, normalized size = 2.04 \[ \frac {1}{4} \, x^{4} \operatorname {arsinh}\left (a x^{2}\right ) + \frac {1}{16} \, a {\left (\frac {\log \left (a + \frac {\sqrt {a^{2} x^{4} + 1}}{x^{2}}\right )}{a^{3}} - \frac {\log \left (-a + \frac {\sqrt {a^{2} x^{4} + 1}}{x^{2}}\right )}{a^{3}} + \frac {2 \, \sqrt {a^{2} x^{4} + 1}}{{\left (a^{4} - \frac {{\left (a^{2} x^{4} + 1\right )} a^{2}}{x^{4}}\right )} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 45, normalized size = 0.90 \[ \frac {x^2\,\mathrm {asinh}\left (a\,x^2\right )\,\left (\frac {x^2}{2}+\frac {1}{4\,a^2\,x^2}\right )}{2}-\frac {x^2\,\sqrt {a^2\,x^4+1}}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 42, normalized size = 0.84 \[ \begin {cases} \frac {x^{4} \operatorname {asinh}{\left (a x^{2} \right )}}{4} - \frac {x^{2} \sqrt {a^{2} x^{4} + 1}}{8 a} + \frac {\operatorname {asinh}{\left (a x^{2} \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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