Optimal. Leaf size=87 \[ \frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {\sinh ^{-1}(a x)}{4 a^3}-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5812, 5783,
5776, 327, 221} \begin {gather*} -\frac {\sinh ^{-1}(a x)^3}{6 a^3}-\frac {\sinh ^{-1}(a x)}{4 a^3}+\frac {x \sqrt {a^2 x^2+1}}{4 a^2}+\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {x^2 \sinh ^{-1}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^2 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}-\frac {\int x \sinh ^{-1}(a x) \, dx}{a}\\ &=-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}+\frac {1}{2} \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}-\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {\sinh ^{-1}(a x)}{4 a^3}-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 72, normalized size = 0.83 \begin {gather*} \frac {3 a x \sqrt {1+a^2 x^2}-3 \left (1+2 a^2 x^2\right ) \sinh ^{-1}(a x)+6 a x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)^3}{12 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.47, size = 69, normalized size = 0.79
method | result | size |
default | \(-\frac {-6 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +6 x^{2} \arcsinh \left (a x \right ) a^{2}+2 \arcsinh \left (a x \right )^{3}-3 \sqrt {a^{2} x^{2}+1}\, a x +3 \arcsinh \left (a x \right )}{12 a^{3}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 102, normalized size = 1.17 \begin {gather*} \frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{12 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 78, normalized size = 0.90 \begin {gather*} \begin {cases} - \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a^{2}} - \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{6 a^{3}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{3}} & \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {x^2\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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