Optimal. Leaf size=176 \[ -\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a} \]
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Rubi [A]
time = 0.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2} a \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\frac {1}{2} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-\frac {1}{2} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\left (2 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}-\left (3 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\left (6 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-3 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{4} \text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-3 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{4} \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+\frac {3}{4} \text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end {align*}
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Mathematica [A]
time = 3.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 5.56, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{a^{6} x^{7} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{3}{\left (a x \right )} - x \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{x\,{\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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