Optimal. Leaf size=66 \[ -\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {15 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.14, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5966, 6034, 5448, 3298} \[ -\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {15 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5966
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+(6 a) \int \frac {x}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh ^5(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {6 \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}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {15 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {15 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 56, normalized size = 0.85 \[ \frac {\frac {1}{\left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)}+\frac {15}{16} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac {3}{4} \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+\frac {3}{16} \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 413, normalized size = 6.26 \[ \frac {3 \, {\left ({\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 4 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 64}{32 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 86, normalized size = 1.30 \[ \frac {-\frac {5}{16 \arctanh \left (a x \right )}-\frac {15 \cosh \left (2 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {15 \Shi \left (2 \arctanh \left (a x \right )\right )}{16}-\frac {3 \cosh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )}+\frac {3 \Shi \left (4 \arctanh \left (a x \right )\right )}{4}-\frac {\cosh \left (6 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {3 \Shi \left (6 \arctanh \left (a x \right )\right )}{16}}{a} \]
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 \[ -12 \, a \int -\frac {x}{{\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - {\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )}\,{d x} + \frac {2}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) - {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^4} \,d x \]
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 \[ \int \frac {1}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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