Optimal. Leaf size=220 \[ \frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}+\frac {6 i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^4}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.41, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6028, 5994, 5952, 4180, 2531, 2282, 6589, 5962, 191} \[ \frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^4}+\frac {6 i \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {6 \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^4} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2282
Rule 2531
Rule 4180
Rule 5952
Rule 5962
Rule 5994
Rule 6028
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^3}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{a^3}\\ &=\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}-\frac {3 \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac {6 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^3}\\ &=-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {6 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {(6 i) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac {(6 i) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}\\ &=-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {6 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}+\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}\\ &=-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {6 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^4}+\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^4}\\ &=-\frac {6 x}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {3 x \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {6 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {\tanh ^{-1}(a x)^3}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{a^4}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}-\frac {6 i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^4}+\frac {6 i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 249, normalized size = 1.13 \[ \frac {\frac {6 i \sqrt {1-a^2 x^2} \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-6 i \sqrt {1-a^2 x^2} \text {Li}_3\left (i e^{-\tanh ^{-1}(a x)}\right )-a^2 x^2 \tanh ^{-1}(a x)^3+3 i \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-3 i \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-6 a x+2 \tanh ^{-1}(a x)^3-3 a x \tanh ^{-1}(a x)^2+6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+6 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-6 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )}{a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{3} \operatorname {artanh}\left (a x\right )^{3}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctanh \left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{3} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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