Optimal. Leaf size=138 \[ \frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {2 \left (1-a^2 x^2\right )}{45 a^2}+\frac {4 \log \left (1-a^2 x^2\right )}{45 a^2}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}+\frac {4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac {8 x \tanh ^{-1}(a x)}{45 a} \]
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Rubi [A] time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5994, 5942, 5910, 260} \[ \frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {2 \left (1-a^2 x^2\right )}{45 a^2}+\frac {4 \log \left (1-a^2 x^2\right )}{45 a^2}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}+\frac {4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac {8 x \tanh ^{-1}(a x)}{45 a} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5910
Rule 5942
Rule 5994
Rubi steps
\begin {align*} \int x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {\int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {4 \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx}{15 a}\\ &=\frac {2 \left (1-a^2 x^2\right )}{45 a^2}+\frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {8 \int \tanh ^{-1}(a x) \, dx}{45 a}\\ &=\frac {2 \left (1-a^2 x^2\right )}{45 a^2}+\frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {8 x \tanh ^{-1}(a x)}{45 a}+\frac {4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}-\frac {8}{45} \int \frac {x}{1-a^2 x^2} \, dx\\ &=\frac {2 \left (1-a^2 x^2\right )}{45 a^2}+\frac {\left (1-a^2 x^2\right )^2}{60 a^2}+\frac {8 x \tanh ^{-1}(a x)}{45 a}+\frac {4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac {x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac {4 \log \left (1-a^2 x^2\right )}{45 a^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.59 \[ \frac {3 a^4 x^4-14 a^2 x^2+16 \log \left (1-a^2 x^2\right )+30 \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^2+4 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \tanh ^{-1}(a x)}{180 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 116, normalized size = 0.84 \[ \frac {6 \, a^{4} x^{4} - 28 \, a^{2} x^{2} + 15 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} - 10 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 32 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 473, normalized size = 3.43 \[ \frac {4}{45} \, a {\left (\frac {2 \, {\left (\frac {10 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {5 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{5} a^{3}}{{\left (a x - 1\right )}^{5}} - \frac {5 \, {\left (a x + 1\right )}^{4} a^{3}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} - \frac {10 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{3}}{a x - 1} - a^{3}} + \frac {30 \, {\left (a x + 1\right )}^{3} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (\frac {{\left (a x + 1\right )}^{6} a^{3}}{{\left (a x - 1\right )}^{6}} - \frac {6 \, {\left (a x + 1\right )}^{5} a^{3}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4} a^{3}}{{\left (a x - 1\right )}^{4}} - \frac {20 \, {\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}^{3}} - \frac {\frac {2 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {7 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1}}{\frac {{\left (a x + 1\right )}^{4} a^{3}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}} - \frac {2 \, \log \left (-\frac {a x + 1}{a x - 1} + 1\right )}{a^{3}} + \frac {2 \, \log \left (-\frac {a x + 1}{a x - 1}\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 219, normalized size = 1.59 \[ \frac {a^{4} \arctanh \left (a x \right )^{2} x^{6}}{6}-\frac {a^{2} \arctanh \left (a x \right )^{2} x^{4}}{2}+\frac {\arctanh \left (a x \right )^{2} x^{2}}{2}+\frac {a^{3} \arctanh \left (a x \right ) x^{5}}{15}-\frac {2 a \arctanh \left (a x \right ) x^{3}}{9}+\frac {x \arctanh \left (a x \right )}{3 a}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{6 a^{2}}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{6 a^{2}}+\frac {\ln \left (a x -1\right )^{2}}{24 a^{2}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{12 a^{2}}+\frac {\ln \left (a x +1\right )^{2}}{24 a^{2}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{12 a^{2}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{12 a^{2}}+\frac {a^{2} x^{4}}{60}-\frac {7 x^{2}}{90}+\frac {4 \ln \left (a x -1\right )}{45 a^{2}}+\frac {4 \ln \left (a x +1\right )}{45 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 93, normalized size = 0.67 \[ \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}}{6 \, a^{2}} + \frac {{\left (3 \, a^{2} x^{4} - 14 \, x^{2} + \frac {16 \, \log \left (a x + 1\right )}{a^{2}} + \frac {16 \, \log \left (a x - 1\right )}{a^{2}}\right )} a + 4 \, {\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x\right )} \operatorname {artanh}\left (a x\right )}{180 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 111, normalized size = 0.80 \[ \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^2}{2}-\frac {{\mathrm {atanh}\left (a\,x\right )}^2}{6\,a^2}-\frac {7\,x^2}{90}+\frac {4\,\ln \left (a^2\,x^2-1\right )}{45\,a^2}+\frac {a^2\,x^4}{60}+\frac {x\,\mathrm {atanh}\left (a\,x\right )}{3\,a}-\frac {2\,a\,x^3\,\mathrm {atanh}\left (a\,x\right )}{9}+\frac {a^3\,x^5\,\mathrm {atanh}\left (a\,x\right )}{15}-\frac {a^2\,x^4\,{\mathrm {atanh}\left (a\,x\right )}^2}{2}+\frac {a^4\,x^6\,{\mathrm {atanh}\left (a\,x\right )}^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 133, normalized size = 0.96 \[ \begin {cases} \frac {a^{4} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}}{6} + \frac {a^{3} x^{5} \operatorname {atanh}{\left (a x \right )}}{15} - \frac {a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{2} + \frac {a^{2} x^{4}}{60} - \frac {2 a x^{3} \operatorname {atanh}{\left (a x \right )}}{9} + \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2} - \frac {7 x^{2}}{90} + \frac {x \operatorname {atanh}{\left (a x \right )}}{3 a} + \frac {8 \log {\left (x - \frac {1}{a} \right )}}{45 a^{2}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{6 a^{2}} + \frac {8 \operatorname {atanh}{\left (a x \right )}}{45 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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