Optimal. Leaf size=168 \[ \frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}-\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a} \]
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Rubi [A]
time = 0.15, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6115, 3393,
3377, 3389, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } \text {Erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}-\frac {\sqrt {\pi } \text {Erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}+\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3377
Rule 3389
Rule 3393
Rule 6115
Rubi steps
\begin {align*} \int \frac {\sqrt {\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\text {Subst}\left (\int \sqrt {x} \cosh ^4(x) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {3 \sqrt {x}}{8}+\frac {1}{2} \sqrt {x} \cosh (2 x)+\frac {1}{8} \sqrt {x} \cosh (4 x)\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\text {Subst}\left (\int \sqrt {x} \cosh (4 x) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {\text {Subst}\left (\int \sqrt {x} \cosh (2 x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a}-\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}-\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a}+\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}-\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}+\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}\\ &=\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a}+\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{64 a}-\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{64 a}+\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{8 a}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{8 a}\\ &=\frac {\tanh ^{-1}(a x)^{3/2}}{4 a}+\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{16 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{4 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{32 a}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 152, normalized size = 0.90 \begin {gather*} \frac {\frac {32 \sqrt {\tanh ^{-1}(a x)} \left (5 a x-3 a^3 x^3+2 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)\right )}{\left (-1+a^2 x^2\right )^2}+\frac {\sqrt {\tanh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \tanh ^{-1}(a x)\right )}{\sqrt {-\tanh ^{-1}(a x)}}+\frac {8 \sqrt {2} \sqrt {\tanh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \tanh ^{-1}(a x)\right )}{\sqrt {-\tanh ^{-1}(a x)}}-8 \sqrt {2} \Gamma \left (\frac {1}{2},2 \tanh ^{-1}(a x)\right )-\Gamma \left (\frac {1}{2},4 \tanh ^{-1}(a x)\right )}{256 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 9.58, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctanh \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {\operatorname {atanh}{\left (a x \right )}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, 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 {\sqrt {\mathrm {atanh}\left (a\,x\right )}}{{\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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