Optimal. Leaf size=42 \[ -\frac {a}{12 x^3}+\frac {a^3}{4 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 x^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6155, 14}
\begin {gather*} \frac {a^3}{4 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 x^4}-\frac {a}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 6155
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x^5} \, dx &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 x^4}+\frac {1}{4} a \int \frac {1-a^2 x^2}{x^4} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 x^4}+\frac {1}{4} a \int \left (\frac {1}{x^4}-\frac {a^2}{x^2}\right ) \, dx\\ &=-\frac {a}{12 x^3}+\frac {a^3}{4 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 x^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 71, normalized size = 1.69 \begin {gather*} -\frac {a}{12 x^3}+\frac {a^3}{4 x}-\frac {\tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)}{2 x^2}+\frac {1}{8} a^4 \log (1-a x)-\frac {1}{8} a^4 \log (1+a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 62, normalized size = 1.48
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {\arctanh \left (a x \right )}{4 a^{4} x^{4}}+\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+\frac {1}{4 a x}-\frac {1}{12 a^{3} x^{3}}-\frac {\ln \left (a x +1\right )}{8}+\frac {\ln \left (a x -1\right )}{8}\right )\) | \(62\) |
default | \(a^{4} \left (-\frac {\arctanh \left (a x \right )}{4 a^{4} x^{4}}+\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+\frac {1}{4 a x}-\frac {1}{12 a^{3} x^{3}}-\frac {\ln \left (a x +1\right )}{8}+\frac {\ln \left (a x -1\right )}{8}\right )\) | \(62\) |
risch | \(\frac {\left (2 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )}{8 x^{4}}-\frac {3 \ln \left (-a x -1\right ) a^{4} x^{4}-3 x^{4} \ln \left (-a x +1\right ) a^{4}-6 a^{3} x^{3}+6 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x -3 \ln \left (-a x +1\right )}{24 x^{4}}\) | \(95\) |
meijerg | \(-\frac {i a^{4} \left (-\frac {i}{3 x^{3} a^{3}}-\frac {i}{x a}+\frac {4 i \left (\frac {3}{8}-\frac {3 a^{4} x^{4}}{8}\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 x^{3} a^{3} \sqrt {a^{2} x^{2}}}\right )}{4}-\frac {i a^{4} \left (\frac {2 i}{x a}+\frac {2 i \left (-a x +1\right ) \left (a x +1\right ) \arctanh \left (a x \right )}{x^{2} a^{2}}\right )}{4}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 61, normalized size = 1.45 \begin {gather*} -\frac {1}{24} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 52, normalized size = 1.24 \begin {gather*} \frac {6 \, a^{3} x^{3} - 2 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 46, normalized size = 1.10 \begin {gather*} - \frac {a^{4} \operatorname {atanh}{\left (a x \right )}}{4} + \frac {a^{3}}{4 x} + \frac {a^{2} \operatorname {atanh}{\left (a x \right )}}{2 x^{2}} - \frac {a}{12 x^{3}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 x^{4}} \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. 160 vs.
\(2 (35) = 70\).
time = 0.40, size = 160, normalized size = 3.81 \begin {gather*} -\frac {1}{3} \, a {\left (\frac {a^{3} {\left (\frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{3} \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 (a x - 1\right )}^{2} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.84, size = 61, normalized size = 1.45 \begin {gather*} \frac {a^3}{4\,x}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,x^4}-\frac {a}{12\,x^3}+\frac {a^5\,\mathrm {atan}\left (\frac {a^2\,x}{\sqrt {-a^2}}\right )}{4\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atanh}\left (a\,x\right )}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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