Optimal. Leaf size=67 \[ 24 x-\frac {24 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4 \]
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Rubi [A]
time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5772, 5798, 8}
\begin {gather*} -\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac {24 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^4+12 x \sinh ^{-1}(a x)^2+24 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5772
Rule 5798
Rubi steps
\begin {align*} \int \sinh ^{-1}(a x)^4 \, dx &=x \sinh ^{-1}(a x)^4-(4 a) \int \frac {x \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4+12 \int \sinh ^{-1}(a x)^2 \, dx\\ &=12 x \sinh ^{-1}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4-(24 a) \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {24 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac {24 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.00 \begin {gather*} 24 x-\frac {24 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 65, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\arcsinh \left (a x \right )^{4} a x -4 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}+12 \arcsinh \left (a x \right )^{2} a x -24 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+24 a x}{a}\) | \(65\) |
default | \(\frac {\arcsinh \left (a x \right )^{4} a x -4 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}+12 \arcsinh \left (a x \right )^{2} a x -24 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}+24 a x}{a}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 73, normalized size = 1.09 \begin {gather*} x \operatorname {arsinh}\left (a x\right )^{4} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}}{a} + 12 \, {\left (\frac {x \operatorname {arsinh}\left (a x\right )^{2}}{a} + \frac {2 \, {\left (x - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a}\right )}}{a}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 112, normalized size = 1.67 \begin {gather*} \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 12 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 4 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 24 \, a x - 24 \, \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 65, normalized size = 0.97 \begin {gather*} \begin {cases} x \operatorname {asinh}^{4}{\left (a x \right )} + 12 x \operatorname {asinh}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a} - \frac {24 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a} & \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 = 0.45, size = 125, normalized size = 1.87 \begin {gather*} x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 4 \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac {3 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a}\right )} a \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 {\mathrm {asinh}\left (a\,x\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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