Optimal. Leaf size=248 \[ -\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8 \tanh ^{-1}(a x)^3}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {8 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {8 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a}+\frac {4 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{5 a} \]
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Rubi [A]
time = 0.19, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6091, 6021,
6131, 6055, 6095, 6205, 6745, 266, 6089} \begin {gather*} -\frac {1-a^2 x^2}{20 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{5 a}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{5 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {8 \tanh ^{-1}(a x)^3}{15 a}-x \tanh ^{-1}(a x)-\frac {8 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6021
Rule 6055
Rule 6089
Rule 6091
Rule 6095
Rule 6131
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3 \, dx &=\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {3}{10} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx+\frac {4}{5} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {1}{5} \int \tanh ^{-1}(a x) \, dx+\frac {8}{15} \int \tanh ^{-1}(a x)^3 \, dx-\frac {4}{5} \int \tanh ^{-1}(a x) \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{5} a \int \frac {x}{1-a^2 x^2} \, dx+\frac {1}{5} (4 a) \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{5} (8 a) \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8 \tanh ^{-1}(a x)^3}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {8}{5} \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8 \tanh ^{-1}(a x)^3}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {8 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}+\frac {16}{5} \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8 \tanh ^{-1}(a x)^3}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {8 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {8 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a}+\frac {8}{5} \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac {1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac {3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac {8 \tanh ^{-1}(a x)^3}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^3+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac {8 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-\frac {8 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a}+\frac {4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{5 a}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 183, normalized size = 0.74 \begin {gather*} \frac {-3+3 a^2 x^2-66 a x \tanh ^{-1}(a x)+6 a^3 x^3 \tanh ^{-1}(a x)+33 \tanh ^{-1}(a x)^2-42 a^2 x^2 \tanh ^{-1}(a x)^2+9 a^4 x^4 \tanh ^{-1}(a x)^2-32 \tanh ^{-1}(a x)^3+60 a x \tanh ^{-1}(a x)^3-40 a^3 x^3 \tanh ^{-1}(a x)^3+12 a^5 x^5 \tanh ^{-1}(a x)^3-96 \tanh ^{-1}(a x)^2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )-30 \log \left (1-a^2 x^2\right )+96 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )}{60 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 30.23, size = 828, normalized size = 3.34
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(828\) |
default | \(\text {Expression too large to display}\) | \(828\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \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 )}^3\,{\left (a^2\,x^2-1\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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