Optimal. Leaf size=70 \[ \frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {x^5 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5665, 3301} \[ \frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {x^5 \sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5665
Rubi steps
\begin {align*} \int \frac {x^5}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \left (\frac {5 \cosh (2 x)}{16 x}-\frac {\cosh (4 x)}{2 x}+\frac {3 \cosh (6 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^6}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^6}\\ &=-\frac {x^5 \sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {5 \text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Chi}\left (4 \sinh ^{-1}(a x)\right )}{2 a^6}+\frac {3 \text {Chi}\left (6 \sinh ^{-1}(a x)\right )}{16 a^6}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 78, normalized size = 1.11 \[ -\frac {-10 \sinh ^{-1}(a x) \text {Chi}\left (2 \sinh ^{-1}(a x)\right )+16 \sinh ^{-1}(a x) \text {Chi}\left (4 \sinh ^{-1}(a x)\right )-6 \sinh ^{-1}(a x) \text {Chi}\left (6 \sinh ^{-1}(a x)\right )+5 \sinh \left (2 \sinh ^{-1}(a x)\right )-4 \sinh \left (4 \sinh ^{-1}(a x)\right )+\sinh \left (6 \sinh ^{-1}(a x)\right )}{32 a^6 \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5}}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
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 = 0.20, size = 78, normalized size = 1.11 \[ \frac {-\frac {5 \sinh \left (2 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {5 \Chi \left (2 \arcsinh \left (a x \right )\right )}{16}+\frac {\sinh \left (4 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )}-\frac {\Chi \left (4 \arcsinh \left (a x \right )\right )}{2}-\frac {\sinh \left (6 \arcsinh \left (a x \right )\right )}{32 \arcsinh \left (a x \right )}+\frac {3 \Chi \left (6 \arcsinh \left (a x \right )\right )}{16}}{a^{6}} \]
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 \[ -\frac {a^{3} x^{8} + a x^{6} + {\left (a^{2} x^{7} + x^{5}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {6 \, a^{5} x^{9} + 12 \, a^{3} x^{7} + 6 \, a x^{5} + 2 \, {\left (3 \, a^{3} x^{7} + 2 \, a x^{5}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (12 \, a^{4} x^{8} + 16 \, a^{2} x^{6} + 5 \, x^{4}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \, {\left (a^{4} x^{3} + a^{2} x\right )} \sqrt {a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{5}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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