Optimal. Leaf size=67 \[ \frac {3 x \sqrt {1+a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1+a^2 x^2}}{16 a}-\frac {3 \sinh ^{-1}(a x)}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 67, 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, 327, 221}
\begin {gather*} -\frac {3 \sinh ^{-1}(a x)}{32 a^4}-\frac {x^3 \sqrt {a^2 x^2+1}}{16 a}+\frac {3 x \sqrt {a^2 x^2+1}}{32 a^3}+\frac {1}{4} x^4 \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \sinh ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{16 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)+\frac {3 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a}\\ &=\frac {3 x \sqrt {1+a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1+a^2 x^2}}{16 a}+\frac {1}{4} x^4 \sinh ^{-1}(a x)-\frac {3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{32 a^3}\\ &=\frac {3 x \sqrt {1+a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1+a^2 x^2}}{16 a}-\frac {3 \sinh ^{-1}(a x)}{32 a^4}+\frac {1}{4} x^4 \sinh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.73 \begin {gather*} \frac {a x \left (3-2 a^2 x^2\right ) \sqrt {1+a^2 x^2}+\left (-3+8 a^4 x^4\right ) \sinh ^{-1}(a x)}{32 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 58, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{16}+\frac {3 a x \sqrt {a^{2} x^{2}+1}}{32}-\frac {3 \arcsinh \left (a x \right )}{32}}{a^{4}}\) | \(58\) |
default | \(\frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{16}+\frac {3 a x \sqrt {a^{2} x^{2}+1}}{32}-\frac {3 \arcsinh \left (a x \right )}{32}}{a^{4}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 59, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {arsinh}\left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} x^{3}}{a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 59, normalized size = 0.88 \begin {gather*} \frac {{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{32 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 61, normalized size = 0.91 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {asinh}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {a^{2} x^{2} + 1}}{16 a} + \frac {3 x \sqrt {a^{2} x^{2} + 1}}{32 a^{3}} - \frac {3 \operatorname {asinh}{\left (a x \right )}}{32 a^{4}} & \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 = -0.01 \begin {gather*} \int x^3\,\mathrm {asinh}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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