Optimal. Leaf size=252 \[ \frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {3 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{6}} \text {Erf}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}-\frac {3 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}-\frac {15 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{6}} \text {Erfi}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a} \]
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Rubi [A]
time = 0.21, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 21, 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 {3 \sqrt {\pi } \text {Erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {Erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{6}} \text {Erf}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}-\frac {3 \sqrt {\pi } \text {Erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}-\frac {15 \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{6}} \text {Erfi}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}+\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 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 )^4} \, dx &=\frac {\text {Subst}\left (\int \sqrt {x} \cosh ^6(x) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {5 \sqrt {x}}{16}+\frac {15}{32} \sqrt {x} \cosh (2 x)+\frac {3}{16} \sqrt {x} \cosh (4 x)+\frac {1}{32} \sqrt {x} \cosh (6 x)\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {\text {Subst}\left (\int \sqrt {x} \cosh (6 x) \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {3 \text {Subst}\left (\int \sqrt {x} \cosh (4 x) \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {15 \text {Subst}\left (\int \sqrt {x} \cosh (2 x) \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}\\ &=\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}-\frac {\text {Subst}\left (\int \frac {\sinh (6 x)}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{384 a}-\frac {3 \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}-\frac {15 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}\\ &=\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}+\frac {\text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{768 a}-\frac {\text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{768 a}+\frac {3 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}-\frac {3 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}+\frac {15 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}-\frac {15 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}\\ &=\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}+\frac {\text {Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{384 a}-\frac {\text {Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{384 a}+\frac {3 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{128 a}-\frac {3 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{128 a}+\frac {15 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{128 a}-\frac {15 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\tanh ^{-1}(a x)}\right )}{128 a}\\ &=\frac {5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{6}} \text {erf}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}-\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\tanh ^{-1}(a x)}\right )}{512 a}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\tanh ^{-1}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{6}} \text {erfi}\left (\sqrt {6} \sqrt {\tanh ^{-1}(a x)}\right )}{768 a}+\frac {15 \sqrt {\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac {3 \sqrt {\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\sqrt {\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 257, normalized size = 1.02 \begin {gather*} \frac {-\frac {3168 x \sqrt {\tanh ^{-1}(a x)}}{\left (-1+a^2 x^2\right )^3}+\frac {3840 a^2 x^3 \sqrt {\tanh ^{-1}(a x)}}{\left (-1+a^2 x^2\right )^3}-\frac {1440 a^4 x^5 \sqrt {\tanh ^{-1}(a x)}}{\left (-1+a^2 x^2\right )^3}+\frac {960 \tanh ^{-1}(a x)^{3/2}}{a}+\frac {\sqrt {6} \sqrt {\tanh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 \tanh ^{-1}(a x)\right )}{a \sqrt {-\tanh ^{-1}(a x)}}+\frac {27 \sqrt {\tanh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \tanh ^{-1}(a x)\right )}{a \sqrt {-\tanh ^{-1}(a x)}}+\frac {135 \sqrt {2} \sqrt {\tanh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \tanh ^{-1}(a x)\right )}{a \sqrt {-\tanh ^{-1}(a x)}}-\frac {135 \sqrt {2} \Gamma \left (\frac {1}{2},2 \tanh ^{-1}(a x)\right )}{a}-\frac {27 \Gamma \left (\frac {1}{2},4 \tanh ^{-1}(a x)\right )}{a}-\frac {\sqrt {6} \Gamma \left (\frac {1}{2},6 \tanh ^{-1}(a x)\right )}{a}}{4608} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 11.58, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctanh \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{4}}\, dx\]
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 )}}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, 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 \frac {\sqrt {\mathrm {atanh}\left (a\,x\right )}}{{\left (a^2\,x^2-1\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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