Optimal. Leaf size=86 \[ \frac {4 x^2}{105 a}-\frac {9 a x^4}{140}+\frac {a^3 x^6}{42}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{105 a^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6159, 6037,
272, 45} \begin {gather*} \frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac {a^3 x^6}{42}-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{105 a^3}-\frac {9 a x^4}{140}+\frac {1}{3} x^3 \tanh ^{-1}(a x)+\frac {4 x^2}{105 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 6037
Rule 6159
Rubi steps
\begin {align*} \int x^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx &=\int \left (x^2 \tanh ^{-1}(a x)-2 a^2 x^4 \tanh ^{-1}(a x)+a^4 x^6 \tanh ^{-1}(a x)\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^4 \tanh ^{-1}(a x) \, dx\right )+a^4 \int x^6 \tanh ^{-1}(a x) \, dx+\int x^2 \tanh ^{-1}(a x) \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2} \, dx+\frac {1}{5} \left (2 a^3\right ) \int \frac {x^5}{1-a^2 x^2} \, dx-\frac {1}{7} a^5 \int \frac {x^7}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )+\frac {1}{5} a^3 \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )-\frac {1}{14} a^5 \text {Subst}\left (\int \frac {x^3}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{5} a^3 \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^5 \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 {4 x^2}{105 a}-\frac {9 a x^4}{140}+\frac {a^3 x^6}{42}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{105 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 86, normalized size = 1.00 \begin {gather*} \frac {4 x^2}{105 a}-\frac {9 a x^4}{140}+\frac {a^3 x^6}{42}+\frac {1}{3} x^3 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{105 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 82, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\arctanh \left (a x \right ) a^{7} x^{7}}{7}-\frac {2 \arctanh \left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{3} x^{3} \arctanh \left (a x \right )}{3}+\frac {a^{6} x^{6}}{42}-\frac {9 a^{4} x^{4}}{140}+\frac {4 a^{2} x^{2}}{105}+\frac {4 \ln \left (a x -1\right )}{105}+\frac {4 \ln \left (a x +1\right )}{105}}{a^{3}}\) | \(82\) |
default | \(\frac {\frac {\arctanh \left (a x \right ) a^{7} x^{7}}{7}-\frac {2 \arctanh \left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{3} x^{3} \arctanh \left (a x \right )}{3}+\frac {a^{6} x^{6}}{42}-\frac {9 a^{4} x^{4}}{140}+\frac {4 a^{2} x^{2}}{105}+\frac {4 \ln \left (a x -1\right )}{105}+\frac {4 \ln \left (a x +1\right )}{105}}{a^{3}}\) | \(82\) |
risch | \(\left (\frac {1}{14} a^{4} x^{7}-\frac {1}{5} a^{2} x^{5}+\frac {1}{6} x^{3}\right ) \ln \left (a x +1\right )-\frac {a^{4} x^{7} \ln \left (-a x +1\right )}{14}+\frac {a^{3} x^{6}}{42}+\frac {a^{2} x^{5} \ln \left (-a x +1\right )}{5}-\frac {9 x^{4} a}{140}-\frac {\ln \left (-a x +1\right ) x^{3}}{6}+\frac {4 x^{2}}{105 a}+\frac {4 \ln \left (a^{2} x^{2}-1\right )}{105 a^{3}}\) | \(110\) |
meijerg | \(\frac {\frac {a^{2} x^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{42}-\frac {2 a^{8} x^{8} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{7}}{4 a^{3}}+\frac {-\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{15}+\frac {2 a^{6} x^{6} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{5}}{2 a^{3}}+\frac {\frac {2 a^{2} x^{2}}{3}-\frac {2 a^{4} x^{4} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}}{4 a^{3}}\) | \(249\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 81, normalized size = 0.94 \begin {gather*} \frac {1}{420} \, a {\left (\frac {10 \, a^{4} x^{6} - 27 \, a^{2} x^{4} + 16 \, x^{2}}{a^{2}} + \frac {16 \, \log \left (a x + 1\right )}{a^{4}} + \frac {16 \, \log \left (a x - 1\right )}{a^{4}}\right )} + \frac {1}{105} \, {\left (15 \, a^{4} x^{7} - 42 \, a^{2} x^{5} + 35 \, x^{3}\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 = 0.35, size = 84, normalized size = 0.98 \begin {gather*} \frac {10 \, a^{6} x^{6} - 27 \, a^{4} x^{4} + 16 \, a^{2} x^{2} + 2 \, {\left (15 \, a^{7} x^{7} - 42 \, a^{5} x^{5} + 35 \, a^{3} x^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 16 \, \log \left (a^{2} x^{2} - 1\right )}{420 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 90, normalized size = 1.05 \begin {gather*} \begin {cases} \frac {a^{4} x^{7} \operatorname {atanh}{\left (a x \right )}}{7} + \frac {a^{3} x^{6}}{42} - \frac {2 a^{2} x^{5} \operatorname {atanh}{\left (a x \right )}}{5} - \frac {9 a x^{4}}{140} + \frac {x^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {4 x^{2}}{105 a} + \frac {8 \log {\left (x - \frac {1}{a} \right )}}{105 a^{3}} + \frac {8 \operatorname {atanh}{\left (a x \right )}}{105 a^{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. 319 vs.
\(2 (72) = 144\).
time = 0.40, size = 319, normalized size = 3.71 \begin {gather*} \frac {4}{105} \, a {\left (\frac {2 \, \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {2 \, \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{4}} - \frac {\frac {2 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} - \frac {11 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {22 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {11 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1}}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{6}} + \frac {2 \, {\left (\frac {70 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {35 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\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^{4} {\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 = 0.95, size = 71, normalized size = 0.83 \begin {gather*} \frac {x^3\,\mathrm {atanh}\left (a\,x\right )}{3}-\frac {9\,a\,x^4}{140}+\frac {4\,\ln \left (a^2\,x^2-1\right )}{105\,a^3}+\frac {4\,x^2}{105\,a}+\frac {a^3\,x^6}{42}-\frac {2\,a^2\,x^5\,\mathrm {atanh}\left (a\,x\right )}{5}+\frac {a^4\,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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