Optimal. Leaf size=68 \[ a^4 x \tanh ^{-1}(a x)-\frac {5}{3} a^3 \log (x)+\frac {2 a^2 \tanh ^{-1}(a x)}{x}+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
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Rubi [A] time = 0.11, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6012, 5910, 260, 5916, 266, 44, 36, 29, 31} \[ \frac {4}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {5}{3} a^3 \log (x)+a^4 x \tanh ^{-1}(a x)+\frac {2 a^2 \tanh ^{-1}(a x)}{x}-\frac {a}{6 x^2}-\frac {\tanh ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 260
Rule 266
Rule 5910
Rule 5916
Rule 6012
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{x^4} \, dx &=\int \left (a^4 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{x^4}-\frac {2 a^2 \tanh ^{-1}(a x)}{x^2}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\right )+a^4 \int \tanh ^{-1}(a x) \, dx+\int \frac {\tanh ^{-1}(a x)}{x^4} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)}{x}+a^4 x \tanh ^{-1}(a x)+\frac {1}{3} a \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx-a^5 \int \frac {x}{1-a^2 x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {2 a^2 \tanh ^{-1}(a x)}{x}+a^4 x \tanh ^{-1}(a x)+\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-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 {2 a^2 \tanh ^{-1}(a x)}{x}+a^4 x \tanh ^{-1}(a x)+\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )+\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 )-a^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-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 {2 a^2 \tanh ^{-1}(a x)}{x}+a^4 x \tanh ^{-1}(a x)-\frac {5}{3} a^3 \log (x)+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 1.00 \[ a^4 x \tanh ^{-1}(a x)-\frac {5}{3} a^3 \log (x)+\frac {2 a^2 \tanh ^{-1}(a x)}{x}+\frac {4}{3} a^3 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 72, normalized size = 1.06 \[ \frac {8 \, a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 10 \, a^{3} x^{3} \log \relax (x) - a x + {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 274, normalized size = 4.03 \[ \frac {1}{3} \, {\left (8 \, a^{2} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right ) - 3 \, a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right ) - 5 \, a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} - 1 \right |}\right ) + {\left (\frac {3 \, a^{2}}{\frac {a x + 1}{a x - 1} - 1} - \frac {\frac {3 \, {\left (a x + 1\right )}^{2} a^{2}}{{\left (a x - 1\right )}^{2}} + \frac {12 \, {\left (a x + 1\right )} a^{2}}{a x - 1} + 5 \, a^{2}}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) + \frac {2 \, {\left (a x + 1\right )} a^{2}}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 69, normalized size = 1.01 \[ a^{4} x \arctanh \left (a x \right )+\frac {2 a^{2} \arctanh \left (a x \right )}{x}-\frac {\arctanh \left (a x \right )}{3 x^{3}}-\frac {a}{6 x^{2}}-\frac {5 a^{3} \ln \left (a x \right )}{3}+\frac {4 a^{3} \ln \left (a x -1\right )}{3}+\frac {4 a^{3} \ln \left (a x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 66, normalized size = 0.97 \[ \frac {1}{6} \, {\left (8 \, a^{2} \log \left (a x + 1\right ) + 8 \, a^{2} \log \left (a x - 1\right ) - 10 \, a^{2} \log \relax (x) - \frac {1}{x^{2}}\right )} a + \frac {1}{3} \, {\left (3 \, a^{4} x + \frac {6 \, a^{2} x^{2} - 1}{x^{3}}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 59, normalized size = 0.87 \[ \frac {4\,a^3\,\ln \left (a^2\,x^2-1\right )}{3}-\frac {a}{6\,x^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{3\,x^3}-\frac {5\,a^3\,\ln \relax (x)}{3}+a^4\,x\,\mathrm {atanh}\left (a\,x\right )+\frac {2\,a^2\,\mathrm {atanh}\left (a\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 75, normalized size = 1.10 \[ \begin {cases} a^{4} x \operatorname {atanh}{\left (a x \right )} - \frac {5 a^{3} \log {\relax (x )}}{3} + \frac {8 a^{3} \log {\left (x - \frac {1}{a} \right )}}{3} + \frac {8 a^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {2 a^{2} \operatorname {atanh}{\left (a x \right )}}{x} - \frac {a}{6 x^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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