Optimal. Leaf size=61 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text{Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac{1}{a x \tanh ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.341975, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx &=a^2 \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, 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 ) \tanh ^{-1}(a x)}-\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\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\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 \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 ) \tanh ^{-1}(a x)}-\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 ) \tanh ^{-1}(a x)}+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 ) \tanh ^{-1}(a x)}+\text{Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end{align*}
Mathematica [A] time = 3.96287, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{{\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) -{\left (a^{3} x^{3} - a x\right )} \log \left (-a x + 1\right )} - \int -\frac{2 \,{\left (3 \, a^{2} x^{2} - 1\right )}}{{\left (a^{5} x^{6} - 2 \, a^{3} x^{4} + a x^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{6} - 2 \, 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{4} x^{5} - 2 \, 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} x^{2} - 1\right )}^{2} x \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]