Optimal. Leaf size=227 \[ -\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {16 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a} \]
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Rubi [A]
time = 0.12, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6091, 6021,
6131, 6055, 2449, 2352, 8, 200} \begin {gather*} -\frac {1}{105} a^4 x^5+\frac {19 a^2 x^3}{315}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}-\frac {16 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {16 \tanh ^{-1}(a x)^2}{35 a}-\frac {32 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}-\frac {38 x}{105} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6091
Rule 6131
Rubi steps
\begin {align*} \int \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2 \, dx &=\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{21} \int \left (1-a^2 x^2\right )^2 \, dx+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{21} \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {3}{35} \int \left (1-a^2 x^2\right ) \, dx+\frac {24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {2 x}{15}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {8 \int 1 \, dx}{35}+\frac {16}{35} \int \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {1}{35} (32 a) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32}{35} \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}+\frac {32}{35} \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {32 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{35 a}\\ &=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac {16 \tanh ^{-1}(a x)^2}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac {32 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {16 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 124, normalized size = 0.55 \begin {gather*} -\frac {114 a x-19 a^3 x^3+3 a^5 x^5+9 (-1+a x)^4 \left (16+29 a x+20 a^2 x^2+5 a^3 x^3\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (-38+57 a^2 x^2-24 a^4 x^4+5 a^6 x^6+96 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )\right )-144 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )}{315 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.68, size = 222, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\arctanh \left (a x \right )^{2} a^{3} x^{3}+\arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \arctanh \left (a x \right )}{35}-\frac {19 a^{2} x^{2} \arctanh \left (a x \right )}{35}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}-\frac {16 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\) | \(222\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\arctanh \left (a x \right )^{2} a^{3} x^{3}+\arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \arctanh \left (a x \right )}{35}-\frac {19 a^{2} x^{2} \arctanh \left (a x \right )}{35}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}-\frac {16 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\) | \(222\) |
risch | \(-\frac {38 x}{105}+\frac {a^{6} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{14}-\frac {1276 \ln \left (a x -1\right )}{3675 a}-\frac {3 a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}+\frac {19 a^{2} x^{3}}{315}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{2}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}-\frac {x \ln \left (-a x +1\right )}{2}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{4}-\frac {19 a \ln \left (a x +1\right ) x^{2}}{70}-\frac {20469}{42875 a}+\frac {\ln \left (-a x +1\right )^{2} x}{4}-\frac {4 \ln \left (-a x +1\right )^{2}}{35 a}-\frac {2453 \ln \left (-a x +1\right )}{7350 a}-\frac {a^{6} \ln \left (-a x +1\right )^{2} x^{7}}{28}+\frac {a^{5} \ln \left (-a x +1\right ) x^{6}}{42}-\frac {a^{6} \ln \left (a x +1\right )^{2} x^{7}}{28}-\frac {a^{5} \ln \left (a x +1\right ) x^{6}}{42}+\frac {\ln \left (a x +1\right )^{2} x}{4}+\frac {4 \ln \left (a x +1\right )^{2}}{35 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {67 \ln \left (a x +1\right )}{210 a}+\frac {3 a^{4} \ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {4 a^{3} \ln \left (a x +1\right ) x^{4}}{35}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{4}+\frac {19 a \ln \left (-a x +1\right ) x^{2}}{70}-\frac {4 a^{3} \ln \left (-a x +1\right ) x^{4}}{35}+\frac {3 a^{4} \ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {a^{4} x^{5}}{105}+\frac {19 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{70 a}-\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a}+\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\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 )}{a}-\frac {16 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}\) | \(509\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 199, normalized size = 0.88 \begin {gather*} -\frac {1}{315} \, a^{2} {\left (\frac {3 \, a^{5} x^{5} - 19 \, a^{3} x^{3} + 114 \, a x + 36 \, \log \left (a x + 1\right )^{2} - 72 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 36 \, \log \left (a x - 1\right )^{2} + 57 \, \log \left (a x - 1\right )}{a^{3}} + \frac {144 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} - \frac {57 \, \log \left (a x + 1\right )}{a^{3}}\right )} - \frac {1}{105} \, {\left (5 \, a^{4} x^{6} - 24 \, a^{2} x^{4} + 57 \, x^{2} - \frac {48 \, \log \left (a x + 1\right )}{a^{2}} - \frac {48 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 35 \, a^{2} x^{3} - 35 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{2}{\left (a x \right )}\right )\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int {\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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