Optimal. Leaf size=103 \[ \frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}-\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a} \]
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Rubi [A] time = 0.144336, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {5956, 6034, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}-\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a} \]
Antiderivative was successfully verified.
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Rule 5956
Rule 6034
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^2} \, dx &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}-\frac{1}{4} a \int \frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\tanh ^{-1}(a x)}} \, dx\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}-\frac{\operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}+\frac{\operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{8 a}-\frac{\operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{8 a}\\ &=\frac{x \sqrt{\tanh ^{-1}(a x)}}{2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^{3/2}}{3 a}+\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}-\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.223862, size = 87, normalized size = 0.84 \[ \sqrt{\tanh ^{-1}(a x)} \left (\frac{\tanh ^{-1}(a x)}{3 a}-\frac{x}{2 \left (a^2 x^2-1\right )}\right )-\frac{\sqrt{\frac{\pi }{2}} \left (\text{Erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )-\text{Erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )\right )}{16 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2}}\sqrt{{\it Artanh} \left ( ax \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{artanh}\left (a x\right )}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{atanh}{\left (a x \right )}}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{artanh}\left (a x\right )}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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