Optimal. Leaf size=141 \[ -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {8 x^3 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)}-\frac {x^3 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac {x \sqrt {a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.28, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5667, 5774, 5665, 3301} \[ -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}-\frac {8 x^3 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)}-\frac {x^3 \sqrt {a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {x \sqrt {a^2 x^2+1}}{a^3 \sinh ^{-1}(a x)}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5665
Rule 5667
Rule 5774
Rubi steps
\begin {align*} \int \frac {x^3}{\sinh ^{-1}(a x)^4} \, dx &=-\frac {x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}+\frac {\int \frac {x^2}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx}{a}+\frac {1}{3} (4 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2}+\frac {8}{3} \int \frac {x^3}{\sinh ^{-1}(a x)^2} \, dx+\frac {\int \frac {x}{\sinh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac {x \sqrt {1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac {8 x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}+\frac {8 \operatorname {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac {x \sqrt {1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac {8 x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {4 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^3}-\frac {x^2}{2 a^2 \sinh ^{-1}(a x)^2}-\frac {2 x^4}{3 \sinh ^{-1}(a x)^2}-\frac {x \sqrt {1+a^2 x^2}}{a^3 \sinh ^{-1}(a x)}-\frac {8 x^3 \sqrt {1+a^2 x^2}}{3 a \sinh ^{-1}(a x)}-\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{3 a^4}+\frac {4 \text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 105, normalized size = 0.74 \[ -\frac {\frac {a x \left (2 a^2 x^2 \sqrt {a^2 x^2+1}+a x \left (4 a^2 x^2+3\right ) \sinh ^{-1}(a x)+2 \sqrt {a^2 x^2+1} \left (8 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2\right )}{\sinh ^{-1}(a x)^3}+2 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )-8 \text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{4}}, x\right ) \]
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 = 0.18, size = 114, normalized size = 0.81 \[ \frac {\frac {\sinh \left (2 \arcsinh \left (a x \right )\right )}{12 \arcsinh \left (a x \right )^{3}}+\frac {\cosh \left (2 \arcsinh \left (a x \right )\right )}{12 \arcsinh \left (a x \right )^{2}}+\frac {\sinh \left (2 \arcsinh \left (a x \right )\right )}{6 \arcsinh \left (a x \right )}-\frac {\Chi \left (2 \arcsinh \left (a x \right )\right )}{3}-\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{24 \arcsinh \left (a x \right )^{3}}-\frac {\cosh \left (4 \arcsinh \left (a x \right )\right )}{12 \arcsinh \left (a x \right )^{2}}-\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{3 \arcsinh \left (a x \right )}+\frac {4 \Chi \left (4 \arcsinh \left (a x \right )\right )}{3}}{a^{4}} \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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