Optimal. Leaf size=58 \[ -\frac {a}{6 x^2}-\frac {\tanh ^{-1}(a x)}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)}{x}-\frac {2}{3} a^3 \log (x)+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right ) \]
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Rubi [A]
time = 0.06, 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 = {6161, 6037,
272, 46, 36, 29, 31} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 6037
Rule 6161
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 \text {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} a^3 \text {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 \text {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 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} a^5 \text {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 = 0.01, size = 58, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 59, normalized size = 1.02
method | result | size |
derivativedivides | \(a^{3} \left (\frac {\arctanh \left (a x \right )}{a x}-\frac {\arctanh \left (a x \right )}{3 a^{3} x^{3}}+\frac {\ln \left (a x -1\right )}{3}+\frac {\ln \left (a x +1\right )}{3}-\frac {1}{6 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}\right )\) | \(59\) |
default | \(a^{3} \left (\frac {\arctanh \left (a x \right )}{a x}-\frac {\arctanh \left (a x \right )}{3 a^{3} x^{3}}+\frac {\ln \left (a x -1\right )}{3}+\frac {\ln \left (a x +1\right )}{3}-\frac {1}{6 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}\right )\) | \(59\) |
risch | \(\frac {\left (3 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )}{6 x^{3}}-\frac {4 \ln \left (x \right ) a^{3} x^{3}-2 \ln \left (-a^{2} x^{2}+1\right ) a^{3} x^{3}+3 x^{2} \ln \left (-a x +1\right ) a^{2}+a x -\ln \left (-a x +1\right )}{6 x^{3}}\) | \(85\) |
meijerg | \(-\frac {a^{3} \left (-\frac {2 \left (10 a^{2} x^{2}+30\right )}{45 a^{2} x^{2}}-\frac {2 \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (i a \right )}{3}+\frac {2}{a^{2} x^{2}}\right )}{4}-\frac {a^{3} \left (\frac {2 \ln \left (1-\sqrt {a^{2} x^{2}}\right )-2 \ln \left (1+\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )+4 \ln \left (x \right )+4 \ln \left (i a \right )\right )}{4}\) | \(175\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 53, normalized size = 0.91 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 64, normalized size = 1.10 \begin {gather*} \frac {2 \, a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{3} x^{3} \log \left (x\right ) - a x + {\left (3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 63, normalized size = 1.09 \begin {gather*} \begin {cases} - \frac {2 a^{3} \log {\left (x \right )}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (50) = 100\).
time = 0.40, size = 204, normalized size = 3.52 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \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 \left (x\right )}{3}+\frac {a^2\,\mathrm {atanh}\left (a\,x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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