Optimal. Leaf size=134 \[ -\frac {1}{36 a \left (1-a^2 x^2\right )^3}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}-\frac {5}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^2}{32 a} \]
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Rubi [A]
time = 0.05, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6107, 6103,
267} \begin {gather*} -\frac {5}{32 a \left (1-a^2 x^2\right )}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}+\frac {5 x \tanh ^{-1}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 \tanh ^{-1}(a x)^2}{32 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6103
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^4} \, dx &=-\frac {1}{36 a \left (1-a^2 x^2\right )^3}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac {1}{36 a \left (1-a^2 x^2\right )^3}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {5}{8} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{36 a \left (1-a^2 x^2\right )^3}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^2}{32 a}-\frac {1}{16} (5 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{36 a \left (1-a^2 x^2\right )^3}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}-\frac {5}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^2}{32 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 81, normalized size = 0.60 \begin {gather*} \frac {68-105 a^2 x^2+45 a^4 x^4-6 a x \left (33-40 a^2 x^2+15 a^4 x^4\right ) \tanh ^{-1}(a x)+45 \left (-1+a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{288 a \left (-1+a^2 x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 222, normalized size = 1.66
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )}{48 \left (a x -1\right )^{3}}+\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {\arctanh \left (a x \right )}{48 \left (a x +1\right )^{3}}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{32}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{288 \left (a x -1\right )^{3}}-\frac {7}{384 \left (a x -1\right )^{2}}+\frac {37}{384 \left (a x -1\right )}-\frac {1}{288 \left (a x +1\right )^{3}}-\frac {7}{384 \left (a x +1\right )^{2}}-\frac {37}{384 \left (a x +1\right )}}{a}\) | \(222\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )}{48 \left (a x -1\right )^{3}}+\frac {\arctanh \left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {\arctanh \left (a x \right )}{48 \left (a x +1\right )^{3}}-\frac {\arctanh \left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {5 \arctanh \left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{32}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{288 \left (a x -1\right )^{3}}-\frac {7}{384 \left (a x -1\right )^{2}}+\frac {37}{384 \left (a x -1\right )}-\frac {1}{288 \left (a x +1\right )^{3}}-\frac {7}{384 \left (a x +1\right )^{2}}-\frac {37}{384 \left (a x +1\right )}}{a}\) | \(222\) |
risch | \(\frac {5 \ln \left (a x +1\right )^{2}}{128 a}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 x^{4} \ln \left (-a x +1\right ) a^{4}-80 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+66 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{192 \left (a^{2} x^{2}-1\right )^{3} a}+\frac {45 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+180 x^{5} \ln \left (-a x +1\right ) a^{5}-135 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+180 a^{4} x^{4}-480 a^{3} x^{3} \ln \left (-a x +1\right )+135 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-420 a^{2} x^{2}+396 a x \ln \left (-a x +1\right )-45 \ln \left (-a x +1\right )^{2}+272}{1152 \left (a^{2} x^{2}-1\right )^{2} a \left (a x -1\right ) \left (a x +1\right )}\) | \(263\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs.
\(2 (114) = 228\).
time = 0.27, size = 240, normalized size = 1.79 \begin {gather*} -\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a}{1152 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 131, normalized size = 0.98 \begin {gather*} \frac {180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} + 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 272}{1152 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 206, normalized size = 1.54 \begin {gather*} \frac {\frac {34}{3\,a}-\frac {35\,a\,x^2}{2}+\frac {15\,a^3\,x^4}{2}}{48\,a^6\,x^6-144\,a^4\,x^4+144\,a^2\,x^2-48}-\ln \left (1-a\,x\right )\,\left (\frac {5\,\ln \left (a\,x+1\right )}{64\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{2\,a^6\,x^6-6\,a^4\,x^4+6\,a^2\,x^2-2}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^2}{128\,a}+\frac {5\,{\ln \left (1-a\,x\right )}^2}{128\,a}-\frac {\ln \left (a\,x+1\right )\,\left (\frac {11\,x}{32\,a}-\frac {5\,a\,x^3}{12}+\frac {5\,a^3\,x^5}{32}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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