Optimal. Leaf size=55 \[ \frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac {5 \log \left (\tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.10, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5968, 3312, 3301} \[ \frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac {5 \log \left (\tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5968
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^6(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {5}{16 x}+\frac {15 \cosh (2 x)}{32 x}+\frac {3 \cosh (4 x)}{16 x}+\frac {\cosh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {5 \log \left (\tanh ^{-1}(a x)\right )}{16 a}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {15 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}\\ &=\frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac {5 \log \left (\tanh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 40, normalized size = 0.73 \[ \frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+6 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )+\text {Chi}\left (6 \tanh ^{-1}(a x)\right )+10 \log \left (\tanh ^{-1}(a x)\right )}{32 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 216, normalized size = 3.93 \[ \frac {20 \, \log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right ) + \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 6 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 6 \, \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 15 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 15 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{64 \, a} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 48, normalized size = 0.87 \[ \frac {15 \Chi \left (2 \arctanh \left (a x \right )\right )}{32 a}+\frac {3 \Chi \left (4 \arctanh \left (a x \right )\right )}{16 a}+\frac {\Chi \left (6 \arctanh \left (a x \right )\right )}{32 a}+\frac {5 \ln \left (\arctanh \left (a x \right )\right )}{16 a} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )}\,{d x} \]
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 \[ \int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^4} \,d x \]
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 \[ \int \frac {1}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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