Optimal. Leaf size=185 \[ -\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-6 \text {PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+6 \text {PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right ) \]
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Rubi [A]
time = 0.29, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6177, 6167,
4267, 2611, 6744, 2320, 6724, 6141, 6109, 197} \begin {gather*} -\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \text {Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+6 \text {Li}_4\left (e^{\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 2320
Rule 2611
Rule 4267
Rule 6109
Rule 6141
Rule 6167
Rule 6177
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-(3 a) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+3 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-(6 a) \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \text {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-6 \text {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \text {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+6 \text {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+6 \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac {6 a x}{\sqrt {1-a^2 x^2}}+\frac {6 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \text {Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+6 \text {Li}_4\left (e^{\tanh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 230, normalized size = 1.24 \begin {gather*} \frac {1}{8} \left (\pi ^4-\frac {48 a x}{\sqrt {1-a^2 x^2}}+\frac {48 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {24 a x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {8 \tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-2 \tanh ^{-1}(a x)^4-8 \tanh ^{-1}(a x)^3 \log \left (1+e^{-\tanh ^{-1}(a x)}\right )+8 \tanh ^{-1}(a x)^3 \log \left (1-e^{\tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+48 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-48 \tanh ^{-1}(a x) \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,-e^{-\tanh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 305, normalized size = 1.65
method | result | size |
default | \(-\frac {\left (\arctanh \left (a x \right )^{3}-3 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-6\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )^{3}+3 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+6\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}+\arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(305\) |
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 \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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