Optimal. Leaf size=122 \[ \frac {4 x \sinh ^{-1}(a x)}{3 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^2}+\frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^4}-\frac {14 \sqrt {a^2 x^2+1}}{9 a^4}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^4}-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a} \]
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Rubi [A] time = 0.22, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5758, 5717, 5653, 261, 5661, 266, 43} \[ \frac {2 \left (a^2 x^2+1\right )^{3/2}}{27 a^4}-\frac {14 \sqrt {a^2 x^2+1}}{9 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^4}+\frac {4 x \sinh ^{-1}(a x)}{3 a^3}-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 5653
Rule 5661
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^3 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}-\frac {2 \int \frac {x \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {2 \int x^2 \sinh ^{-1}(a x) \, dx}{3 a}\\ &=-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac {2}{9} \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx+\frac {4 \int \sinh ^{-1}(a x) \, dx}{3 a^3}\\ &=\frac {4 x \sinh ^{-1}(a x)}{3 a^3}-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}\\ &=-\frac {4 \sqrt {1+a^2 x^2}}{3 a^4}+\frac {4 x \sinh ^{-1}(a x)}{3 a^3}-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}+\frac {1}{9} \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {14 \sqrt {1+a^2 x^2}}{9 a^4}+\frac {2 \left (1+a^2 x^2\right )^{3/2}}{27 a^4}+\frac {4 x \sinh ^{-1}(a x)}{3 a^3}-\frac {2 x^3 \sinh ^{-1}(a x)}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.65 \[ \frac {2 \left (a^2 x^2-20\right ) \sqrt {a^2 x^2+1}+9 \left (a^2 x^2-2\right ) \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2-6 a x \left (a^2 x^2-6\right ) \sinh ^{-1}(a x)}{27 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 98, normalized size = 0.80 \[ \frac {9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )}}{27 \, a^{4}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 113, normalized size = 0.93 \[ \frac {9 \arcsinh \left (a x \right )^{2} x^{4} a^{4}-9 \arcsinh \left (a x \right )^{2} a^{2} x^{2}-6 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+2 x^{4} a^{4}-38 a^{2} x^{2}-18 \arcsinh \left (a x \right )^{2}+36 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -40}{27 a^{4} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 101, normalized size = 0.83 \[ \frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{2} + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} - \frac {2 \, {\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )}{9 \, a^{3}} \]
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 \[ \int \frac {x^3\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.17, size = 121, normalized size = 0.99 \[ \begin {cases} - \frac {2 x^{3} \operatorname {asinh}{\left (a x \right )}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1}}{27 a^{2}} + \frac {4 x \operatorname {asinh}{\left (a x \right )}}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a^{4}} - \frac {40 \sqrt {a^{2} x^{2} + 1}}{27 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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