Optimal. Leaf size=98 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2} \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{2} \text{Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac{1}{a x \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.650164, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]
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 \operatorname{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+\operatorname{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 \operatorname{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}+\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\operatorname{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 )+\operatorname{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} \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac{1}{2} \operatorname{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}-\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\operatorname{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} \operatorname{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*}
Mathematica [A] time = 4.26807, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.249, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (-a x + 1\right )} + \int -\frac{2 \,{\left (5 \, a^{2} x^{2} - 1\right )}}{{\left (a^{7} x^{8} - 3 \, a^{5} x^{6} + 3 \, a^{3} x^{4} - a x^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{7} x^{8} - 3 \, a^{5} x^{6} + 3 \, a^{3} x^{4} - a x^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname{artanh}\left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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