Optimal. Leaf size=56 \[ -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^4}-\frac {x^3 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5665, 3301} \[ -\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^4}-\frac {x^3 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5665
Rubi steps
\begin {align*} \int \frac {x^3}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^3 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\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 )}{a^4}\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^4}\\ &=-\frac {x^3 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^4}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.00 \[ -\frac {4 \sinh ^{-1}(a x) \text {Chi}\left (2 \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x) \text {Chi}\left (4 \sinh ^{-1}(a x)\right )-2 \sinh \left (2 \sinh ^{-1}(a x)\right )+\sinh \left (4 \sinh ^{-1}(a x)\right )}{8 a^4 \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
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 = 0.16, size = 54, normalized size = 0.96 \[ \frac {\frac {\sinh \left (2 \arcsinh \left (a x \right )\right )}{4 \arcsinh \left (a x \right )}-\frac {\Chi \left (2 \arcsinh \left (a x \right )\right )}{2}-\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )}+\frac {\Chi \left (4 \arcsinh \left (a x \right )\right )}{2}}{a^{4}} \]
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 \[ -\frac {a^{3} x^{6} + a x^{4} + {\left (a^{2} x^{5} + x^{3}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {4 \, a^{5} x^{7} + 8 \, a^{3} x^{5} + 4 \, a x^{3} + 2 \, {\left (2 \, a^{3} x^{5} + a x^{3}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (8 \, a^{4} x^{6} + 10 \, a^{2} x^{4} + 3 \, x^{2}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} + a^{2} x\right )} \sqrt {a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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