Optimal. Leaf size=58 \[ -\frac {2}{3} a^3 \log (x)+\frac {a^2 \tanh ^{-1}(a x)}{x}+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6014, 5916, 266, 44, 36, 29, 31} \[ \frac {1}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {2}{3} a^3 \log (x)+\frac {a^2 \tanh ^{-1}(a x)}{x}-\frac {a}{6 x^2}-\frac {\tanh ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 5916
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x^4} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\right )+\int \frac {\tanh ^{-1}(a x)}{x^4} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)}{x}+\frac {1}{3} a \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx-a^3 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)}{x}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)}{x}+\frac {1}{6} a \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a}{6 x^2}-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)}{x}-\frac {2}{3} a^3 \log (x)+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.00 \[ -\frac {2}{3} a^3 \log (x)+\frac {a^2 \tanh ^{-1}(a x)}{x}+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 64, normalized size = 1.10 \[ \frac {2 \, a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{3} x^{3} \log \relax (x) - a x + {\left (3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 204, normalized size = 3.52 \[ \frac {2}{3} \, {\left (a^{2} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right ) - a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} - 1 \right |}\right ) + \frac {{\left (a x + 1\right )} a^{2}}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}} - \frac {{\left (\frac {3 \, {\left (a x + 1\right )} a^{2}}{a x - 1} + a^{2}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 59, normalized size = 1.02 \[ \frac {a^{2} \arctanh \left (a x \right )}{x}-\frac {\arctanh \left (a x \right )}{3 x^{3}}-\frac {a}{6 x^{2}}-\frac {2 a^{3} \ln \left (a x \right )}{3}+\frac {a^{3} \ln \left (a x -1\right )}{3}+\frac {a^{3} \ln \left (a x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 53, normalized size = 0.91 \[ \frac {1}{6} \, {\left (2 \, a^{2} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a + \frac {{\left (3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 49, normalized size = 0.84 \[ \frac {a^3\,\ln \left (a^2\,x^2-1\right )}{3}-\frac {a}{6\,x^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{3\,x^3}-\frac {2\,a^3\,\ln \relax (x)}{3}+\frac {a^2\,\mathrm {atanh}\left (a\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.19, size = 63, normalized size = 1.09 \[ \begin {cases} - \frac {2 a^{3} \log {\relax (x )}}{3} + \frac {2 a^{3} \log {\left (x - \frac {1}{a} \right )}}{3} + \frac {2 a^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {a^{2} \operatorname {atanh}{\left (a x \right )}}{x} - \frac {a}{6 x^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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