Optimal. Leaf size=120 \[ -\frac{1}{b x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac{3}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0799972, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2163, 2162} \[ -\frac{1}{b x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac{3}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2163
Rule 2162
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}-\frac{3 \int \frac{1}{x^{5/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=-\frac{1}{b x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \int \frac{1}{x^{3/2} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac{3}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac{1}{b x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}+\frac{(3 b) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac{3}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}-\frac{1}{b x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac{1}{b x^{3/2} \tanh ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.104345, size = 104, normalized size = 0.87 \[ -\frac{b \sqrt{x}}{\tanh ^{-1}(\tanh (a+b x)) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}-\frac{2}{\sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.124, size = 105, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}\sqrt{x}}}-{\frac{b}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }\sqrt{x}}-3\,{\frac{b}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10007, size = 323, normalized size = 2.69 \begin{align*} \left [\frac{3 \,{\left (b x^{2} + a x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 2 \,{\left (3 \, b x + 2 \, a\right )} \sqrt{x}}{2 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac{3 \,{\left (b x^{2} + a x\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (3 \, b x + 2 \, a\right )} \sqrt{x}}{a^{2} b x^{2} + a^{3} x}\right ] \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{1}{x^{\frac{3}{2}} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15267, size = 66, normalized size = 0.55 \begin{align*} -\frac{3 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{3 \, b x + 2 \, a}{{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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