Optimal. Leaf size=338 \[ -\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a}+\frac {24 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{35 a} \]
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Rubi [A]
time = 0.25, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps
used = 17, 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 {\left (1-a^2 x^2\right )^2}{140 a}-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac {24 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{35 a}-\frac {48 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {16 \tanh ^{-1}(a x)^3}{35 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {48 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{35 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 )^3 \tanh ^{-1}(a x)^3 \, dx &=\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {1}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {4}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx-\frac {9}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx+\frac {24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {8}{105} \int \tanh ^{-1}(a x) \, dx-\frac {6}{35} \int \tanh ^{-1}(a x) \, dx+\frac {16}{35} \int \tanh ^{-1}(a x)^3 \, dx-\frac {24}{35} \int \tanh ^{-1}(a x) \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac {1}{105} (8 a) \int \frac {x}{1-a^2 x^2} \, dx+\frac {1}{35} (6 a) \int \frac {x}{1-a^2 x^2} \, dx+\frac {1}{35} (24 a) \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{35} (48 a) \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48}{35} \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}+\frac {96}{35} \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {48}{35} \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \tanh ^{-1}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac {12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac {16 \tanh ^{-1}(a x)^3}{35 a}+\frac {16}{35} x \tanh ^{-1}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac {48 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a}+\frac {24 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{35 a}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 231, normalized size = 0.68 \begin {gather*} -\frac {29-32 a^2 x^2+3 a^4 x^4+456 a x \tanh ^{-1}(a x)-76 a^3 x^3 \tanh ^{-1}(a x)+12 a^5 x^5 \tanh ^{-1}(a x)-228 \tanh ^{-1}(a x)^2+342 a^2 x^2 \tanh ^{-1}(a x)^2-144 a^4 x^4 \tanh ^{-1}(a x)^2+30 a^6 x^6 \tanh ^{-1}(a x)^2+192 \tanh ^{-1}(a x)^3-420 a x \tanh ^{-1}(a x)^3+420 a^3 x^3 \tanh ^{-1}(a x)^3-252 a^5 x^5 \tanh ^{-1}(a x)^3+60 a^7 x^7 \tanh ^{-1}(a x)^3+576 \tanh ^{-1}(a x)^2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+196 \log \left (1-a^2 x^2\right )-576 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-288 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )}{420 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 = 49.18, size = 978, normalized size = 2.89
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(978\) |
default | \(\text {Expression too large to display}\) | \(978\) |
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 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{3}{\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 )}^3\,{\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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