Optimal. Leaf size=99 \[ -\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a} \]
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Rubi [A]
time = 0.45, 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)^2} \, 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)^2} \, dx &=a^2 \int \frac {x}{\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 )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+a \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (3 a^3\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}+a \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+a^3 \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\text {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \text {Subst}\left (\int \left (-\frac {1}{8 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}+\text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\text {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{8} \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {3}{8} \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}-\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \tanh ^{-1}(a x)\right )+2 \left (\frac {1}{2} \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end {align*}
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Mathematica [A]
time = 3.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 5.47, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )^{2}}\, dx\]
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}^{2}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \operatorname {atanh}^{2}{\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 )}^2\,{\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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