Optimal. Leaf size=105 \[ -\frac {\sinh ^{-1}(a x)^4}{8 a^3}-\frac {3 \sinh ^{-1}(a x)^2}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 a^2}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 x^2}{8 a}-\frac {3 x^2 \sinh ^{-1}(a x)^2}{4 a} \]
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Rubi [A] time = 0.22, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5758, 5675, 5661, 30} \[ \frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 a^2}+\frac {3 x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac {\sinh ^{-1}(a x)^4}{8 a^3}-\frac {3 \sinh ^{-1}(a x)^2}{8 a^3}-\frac {3 x^2}{8 a}-\frac {3 x^2 \sinh ^{-1}(a x)^2}{4 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5661
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^2 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 a^2}-\frac {\int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}-\frac {3 \int x \sinh ^{-1}(a x)^2 \, dx}{2 a}\\ &=-\frac {3 x^2 \sinh ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 a^2}-\frac {\sinh ^{-1}(a x)^4}{8 a^3}+\frac {3}{2} \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 x^2 \sinh ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 a^2}-\frac {\sinh ^{-1}(a x)^4}{8 a^3}-\frac {3 \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}-\frac {3 \int x \, dx}{4 a}\\ &=-\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{4 a^2}-\frac {3 \sinh ^{-1}(a x)^2}{8 a^3}-\frac {3 x^2 \sinh ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 a^2}-\frac {\sinh ^{-1}(a x)^4}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.79 \[ -\frac {3 a^2 x^2-4 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3+\left (6 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2-6 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^4}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 128, normalized size = 1.22 \[ \frac {4 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 3 \, a^{2} x^{2} - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{8 \, a^{3}} \]
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 \[ \int \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 84, normalized size = 0.80 \[ -\frac {-4 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x +6 \arcsinh \left (a x \right )^{2} a^{2} x^{2}+\arcsinh \left (a x \right )^{4}-6 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +3 a^{2} x^{2}+3 \arcsinh \left (a x \right )^{2}+3}{8 a^{3}} \]
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 \[ \int \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.20, size = 100, normalized size = 0.95 \[ \begin {cases} - \frac {3 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x^{2}}{8 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{2 a^{2}} + \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{4 a^{2}} - \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{8 a^{3}} - \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{8 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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