Optimal. Leaf size=50 \[ -\frac {x^2 \sqrt {1+a^2 x^4}}{8 a}+\frac {\sinh ^{-1}\left (a x^2\right )}{8 a^2}+\frac {1}{4} x^4 \sinh ^{-1}\left (a x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {5875, 12, 281,
327, 221} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 281
Rule 327
Rule 5875
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 \text {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 {\text {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 = 0.01, size = 44, normalized size = 0.88 \begin {gather*} \frac {-a x^2 \sqrt {1+a^2 x^4}+\left (1+2 a^2 x^4\right ) \sinh ^{-1}\left (a x^2\right )}{8 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 71, normalized size = 1.42
method | result | size |
default | \(\frac {x^{4} \arcsinh \left (a \,x^{2}\right )}{4}-\frac {a \left (\frac {x^{2} \sqrt {a^{2} x^{4}+1}}{4 a^{2}}-\frac {\ln \left (\frac {a^{2} x^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{4}+1}\right )}{4 a^{2} \sqrt {a^{2}}}\right )}{2}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (42) = 84\).
time = 0.27, size = 102, normalized size = 2.04 \begin {gather*} \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 )} \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 = 1.04 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 42, normalized size = 0.84 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 74, normalized size = 1.48 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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