Optimal. Leaf size=43 \[ \frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^6} \]
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Rubi [A] time = 0.14, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6034, 5448, 3298} \[ \frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^6} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 6034
Rubi steps
\begin {align*} \int \frac {x^5}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 x}-\frac {\sinh (4 x)}{8 x}+\frac {\sinh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^6}+\frac {5 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^6}\\ &=\frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^6}-\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^6}+\frac {\text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^6}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 33, normalized size = 0.77 \[ \frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )-4 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+\text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 200, normalized size = 4.65 \[ \frac {\operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) - 4 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 5 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 5 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{64 \, a^{6}} \]
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 \[ \int \frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 33, normalized size = 0.77 \[ \frac {-\frac {\Shi \left (4 \arctanh \left (a x \right )\right )}{8}+\frac {\Shi \left (6 \arctanh \left (a x \right )\right )}{32}+\frac {5 \Shi \left (2 \arctanh \left (a x \right )\right )}{32}}{a^{6}} \]
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 \[ \int \frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )}\,{d x} \]
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 \[ \int \frac {x^5}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^4} \,d x \]
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 \[ \int \frac {x^{5}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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