Optimal. Leaf size=135 \[ \frac{7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac{7 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}{b^{9/2}}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{7 x^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.103377, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2159, 2162} \[ \frac{7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac{7 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}{b^{9/2}}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{7 x^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2162
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{7 \int \frac{x^{5/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=\frac{7 x^{5/2}}{5 b^2}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}-\frac{\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^2}\\ &=\frac{7 x^{5/2}}{5 b^2}+\frac{7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^3}\\ &=\frac{7 x^{5/2}}{5 b^2}+\frac{7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac{7 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}-\frac{\left (7 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3\right ) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^4}\\ &=\frac{7 x^{5/2}}{5 b^2}+\frac{7 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^3}+\frac{7 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac{7 \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}}{b^{9/2}}-\frac{x^{7/2}}{b \tanh ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.186507, size = 144, normalized size = 1.07 \[ -\frac{4 x^{3/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{3 b^3}+\frac{\sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3}{b^4 \tanh ^{-1}(\tanh (a+b x))}+\frac{6 \sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}{b^4}-\frac{7 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{9/2}}+\frac{2 x^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.136, size = 452, normalized size = 3.4 \begin{align*} \text{result too large to display} \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 = 2.10656, size = 427, normalized size = 3.16 \begin{align*} \left [\frac{105 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt{x}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{105 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} \sqrt{x}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1208, size = 103, normalized size = 0.76 \begin{align*} -\frac{7 \, a^{3} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{a^{3} \sqrt{x}}{{\left (b x + a\right )} b^{4}} + \frac{2 \,{\left (3 \, b^{8} x^{\frac{5}{2}} - 10 \, a b^{7} x^{\frac{3}{2}} + 45 \, a^{2} b^{6} \sqrt{x}\right )}}{15 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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