Optimal. Leaf size=89 \[ -\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+\frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a}+\frac {9 \text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.39, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5966, 6032, 6034, 5448, 3301, 5968, 3312} \[ -\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+\frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a}+\frac {9 \text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5448
Rule 5966
Rule 5968
Rule 6032
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+(3 a) \int \frac {x}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+3 \int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx+\left (15 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^6(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {15 \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {3 \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 {15 \operatorname {Subst}\left (\int \left (-\frac {1}{16 x}-\frac {\cosh (2 x)}{32 x}+\frac {\cosh (4 x)}{16 x}+\frac {\cosh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}-\frac {15 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {15 \operatorname {Subst}\left (\int \frac {\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {9 \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 (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {45 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{16 a}+\frac {3 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a}+\frac {9 \text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 83, normalized size = 0.93 \[ \frac {1}{16} \left (\frac {48 x}{\left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)}+\frac {8}{a \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^2}+\frac {15 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {24 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{a}+\frac {9 \text {Chi}\left (6 \tanh ^{-1}(a x)\right )}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 435, normalized size = 4.89 \[ \frac {192 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + 3 \, {\left (3 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 3 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 8 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 8 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 64}{32 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 131, normalized size = 1.47 \[ \frac {-\frac {5}{32 \arctanh \left (a x \right )^{2}}-\frac {15 \cosh \left (2 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {15 \sinh \left (2 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {15 \Chi \left (2 \arctanh \left (a x \right )\right )}{16}-\frac {3 \cosh \left (4 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )^{2}}-\frac {3 \sinh \left (4 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )}+\frac {3 \Chi \left (4 \arctanh \left (a x \right )\right )}{2}-\frac {\cosh \left (6 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {3 \sinh \left (6 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {9 \Chi \left (6 \arctanh \left (a x \right )\right )}{16}}{a} \]
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 \[ \frac {2 \, {\left (3 \, a x \log \left (a x + 1\right ) - 3 \, a x \log \left (-a x + 1\right ) + 1\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )^{2}} - \int -\frac {6 \, {\left (5 \, a^{2} x^{2} + 1\right )}}{{\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - {\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^4} \,d x \]
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 \[ \int \frac {1}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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