Optimal. Leaf size=72 \[ \frac {x^2}{35 a^3}+\frac {x^4}{70 a}-\frac {a x^6}{42}+\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{35 a^5} \]
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Rubi [A]
time = 0.08, antiderivative size = 72, 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,
272, 45} \begin {gather*} \frac {x^2}{35 a^3}-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{35 a^5}-\frac {a x^6}{42}+\frac {1}{5} x^5 \tanh ^{-1}(a x)+\frac {x^4}{70 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6037
Rule 6161
Rubi steps
\begin {align*} \int x^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^6 \tanh ^{-1}(a x) \, dx\right )+\int x^4 \tanh ^{-1}(a x) \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac {1}{5} a \int \frac {x^5}{1-a^2 x^2} \, dx+\frac {1}{7} a^3 \int \frac {x^7}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac {1}{14} a^3 \text {Subst}\left (\int \frac {x^3}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)-\frac {1}{10} a \text {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{14} a^3 \text {Subst}\left (\int \left (-\frac {1}{a^6}-\frac {x}{a^4}-\frac {x^2}{a^2}-\frac {1}{a^6 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{35 a^3}+\frac {x^4}{70 a}-\frac {a x^6}{42}+\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{35 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 72, normalized size = 1.00 \begin {gather*} \frac {x^2}{35 a^3}+\frac {x^4}{70 a}-\frac {a x^6}{42}+\frac {1}{5} x^5 \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{35 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 70, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right ) a^{7} x^{7}}{7}+\frac {\arctanh \left (a x \right ) a^{5} x^{5}}{5}-\frac {a^{6} x^{6}}{42}+\frac {a^{4} x^{4}}{70}+\frac {a^{2} x^{2}}{35}+\frac {\ln \left (a x -1\right )}{35}+\frac {\ln \left (a x +1\right )}{35}}{a^{5}}\) | \(70\) |
default | \(\frac {-\frac {\arctanh \left (a x \right ) a^{7} x^{7}}{7}+\frac {\arctanh \left (a x \right ) a^{5} x^{5}}{5}-\frac {a^{6} x^{6}}{42}+\frac {a^{4} x^{4}}{70}+\frac {a^{2} x^{2}}{35}+\frac {\ln \left (a x -1\right )}{35}+\frac {\ln \left (a x +1\right )}{35}}{a^{5}}\) | \(70\) |
risch | \(\left (-\frac {1}{14} a^{2} x^{7}+\frac {1}{10} x^{5}\right ) \ln \left (a x +1\right )+\frac {a^{2} x^{7} \ln \left (-a x +1\right )}{14}-\frac {x^{6} a}{42}-\frac {x^{5} \ln \left (-a x +1\right )}{10}+\frac {x^{4}}{70 a}+\frac {x^{2}}{35 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{35 a^{5}}\) | \(87\) |
meijerg | \(-\frac {\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{42}-\frac {2 x^{8} a^{8} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{7}}{4 a^{5}}-\frac {-\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{15}+\frac {2 a^{6} x^{6} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{5}}{4 a^{5}}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 73, normalized size = 1.01 \begin {gather*} -\frac {1}{210} \, a {\left (\frac {5 \, a^{4} x^{6} - 3 \, a^{2} x^{4} - 6 \, x^{2}}{a^{4}} - \frac {6 \, \log \left (a x + 1\right )}{a^{6}} - \frac {6 \, \log \left (a x - 1\right )}{a^{6}}\right )} - \frac {1}{35} \, {\left (5 \, a^{2} x^{7} - 7 \, x^{5}\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 = 0.37, size = 76, normalized size = 1.06 \begin {gather*} -\frac {5 \, a^{6} x^{6} - 3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} + 3 \, {\left (5 \, a^{7} x^{7} - 7 \, a^{5} x^{5}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 6 \, \log \left (a^{2} x^{2} - 1\right )}{210 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 71, normalized size = 0.99 \begin {gather*} \begin {cases} - \frac {a^{2} x^{7} \operatorname {atanh}{\left (a x \right )}}{7} - \frac {a x^{6}}{42} + \frac {x^{5} \operatorname {atanh}{\left (a x \right )}}{5} + \frac {x^{4}}{70 a} + \frac {x^{2}}{35 a^{3}} + \frac {2 \log {\left (x - \frac {1}{a} \right )}}{35 a^{5}} + \frac {2 \operatorname {atanh}{\left (a x \right )}}{35 a^{5}} & \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. 335 vs.
\(2 (60) = 120\).
time = 0.40, size = 335, normalized size = 4.65 \begin {gather*} \frac {2}{105} \, a {\left (\frac {3 \, \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{6}} - \frac {3 \, \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{6}} - \frac {\frac {3 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {36 \, {\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 {36 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1}}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{6}} - \frac {3 \, {\left (\frac {35 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {35 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {70 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {14 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )}}{a x - 1} - 1\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^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{7}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 61, normalized size = 0.85 \begin {gather*} \frac {\frac {\ln \left (a^2\,x^2-1\right )}{35}+\frac {a^2\,x^2}{35}+\frac {a^4\,x^4}{70}}{a^5}-\frac {a\,x^6}{42}+\frac {x^5\,\mathrm {atanh}\left (a\,x\right )}{5}-\frac {a^2\,x^7\,\mathrm {atanh}\left (a\,x\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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