Optimal. Leaf size=63 \[ \frac {x}{12 a^3}+\frac {x^3}{36 a}-\frac {a x^5}{30}-\frac {\tanh ^{-1}(a x)}{12 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x) \]
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Rubi [A]
time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6161, 6037,
308, 212} \begin {gather*} -\frac {\tanh ^{-1}(a x)}{12 a^4}+\frac {x}{12 a^3}-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac {a x^5}{30}+\frac {1}{4} x^4 \tanh ^{-1}(a x)+\frac {x^3}{36 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 6037
Rule 6161
Rubi steps
\begin {align*} \int x^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^5 \tanh ^{-1}(a x) \, dx\right )+\int x^3 \tanh ^{-1}(a x) \, dx\\ &=\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {1}{6} a^3 \int \frac {x^6}{1-a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)-\frac {1}{4} a \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx+\frac {1}{6} a^3 \int \left (-\frac {1}{a^6}-\frac {x^2}{a^4}-\frac {x^4}{a^2}+\frac {1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {x}{12 a^3}+\frac {x^3}{36 a}-\frac {a x^5}{30}+\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{6 a^3}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^3}\\ &=\frac {x}{12 a^3}+\frac {x^3}{36 a}-\frac {a x^5}{30}-\frac {\tanh ^{-1}(a x)}{12 a^4}+\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 79, normalized size = 1.25 \begin {gather*} \frac {x}{12 a^3}+\frac {x^3}{36 a}-\frac {a x^5}{30}+\frac {1}{4} x^4 \tanh ^{-1}(a x)-\frac {1}{6} a^2 x^6 \tanh ^{-1}(a x)+\frac {\log (1-a x)}{24 a^4}-\frac {\log (1+a x)}{24 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 66, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{6}+\frac {a^{4} x^{4} \arctanh \left (a x \right )}{4}-\frac {a^{5} x^{5}}{30}+\frac {a^{3} x^{3}}{36}+\frac {a x}{12}+\frac {\ln \left (a x -1\right )}{24}-\frac {\ln \left (a x +1\right )}{24}}{a^{4}}\) | \(66\) |
default | \(\frac {-\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{6}+\frac {a^{4} x^{4} \arctanh \left (a x \right )}{4}-\frac {a^{5} x^{5}}{30}+\frac {a^{3} x^{3}}{36}+\frac {a x}{12}+\frac {\ln \left (a x -1\right )}{24}-\frac {\ln \left (a x +1\right )}{24}}{a^{4}}\) | \(66\) |
risch | \(\left (-\frac {1}{12} x^{6} a^{2}+\frac {1}{8} x^{4}\right ) \ln \left (a x +1\right )+\frac {a^{2} x^{6} \ln \left (-a x +1\right )}{12}-\frac {a \,x^{5}}{30}-\frac {x^{4} \ln \left (-a x +1\right )}{8}+\frac {x^{3}}{36 a}+\frac {x}{12 a^{3}}+\frac {\ln \left (-a x +1\right )}{24 a^{4}}-\frac {\ln \left (a x +1\right )}{24 a^{4}}\) | \(93\) |
meijerg | \(-\frac {i \left (-\frac {2 i x a \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{315}-\frac {i x a \left (-7 a^{6} x^{6}+7\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}-\frac {i \left (\frac {i x a \left (5 a^{2} x^{2}+15\right )}{15}+\frac {i x a \left (-5 a^{4} x^{4}+5\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{10 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 72, normalized size = 1.14 \begin {gather*} -\frac {1}{360} \, a {\left (\frac {2 \, {\left (6 \, a^{4} x^{5} - 5 \, a^{2} x^{3} - 15 \, x\right )}}{a^{4}} + \frac {15 \, \log \left (a x + 1\right )}{a^{5}} - \frac {15 \, \log \left (a x - 1\right )}{a^{5}}\right )} - \frac {1}{12} \, {\left (2 \, a^{2} x^{6} - 3 \, x^{4}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 61, normalized size = 0.97 \begin {gather*} -\frac {12 \, a^{5} x^{5} - 10 \, a^{3} x^{3} - 30 \, a x + 15 \, {\left (2 \, a^{6} x^{6} - 3 \, a^{4} x^{4} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{360 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 54, normalized size = 0.86 \begin {gather*} \begin {cases} - \frac {a^{2} x^{6} \operatorname {atanh}{\left (a x \right )}}{6} - \frac {a x^{5}}{30} + \frac {x^{4} \operatorname {atanh}{\left (a x \right )}}{4} + \frac {x^{3}}{36 a} + \frac {x}{12 a^{3}} - \frac {\operatorname {atanh}{\left (a x \right )}}{12 a^{4}} & \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. 227 vs.
\(2 (51) = 102\).
time = 0.41, size = 227, normalized size = 3.60 \begin {gather*} -\frac {1}{45} \, a {\left (\frac {\frac {45 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {25 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {35 \, {\left (a x + 1\right )}}{a x - 1} - 7}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{5}} + \frac {30 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {2 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{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 )}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 51, normalized size = 0.81 \begin {gather*} \frac {\frac {a\,x}{12}-\frac {\mathrm {atanh}\left (a\,x\right )}{12}+\frac {a^3\,x^3}{36}}{a^4}-\frac {a\,x^5}{30}+\frac {x^4\,\mathrm {atanh}\left (a\,x\right )}{4}-\frac {a^2\,x^6\,\mathrm {atanh}\left (a\,x\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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