Optimal. Leaf size=143 \[ \frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{2 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{3 b^3}+\frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{b^4}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{7/2}}{b^{9/2}}+\frac{2 x^{7/2}}{7 b} \]
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Rubi [A] time = 0.127289, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2159, 2162} \[ \frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{2 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{3 b^3}+\frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{b^4}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{7/2}}{b^{9/2}}+\frac{2 x^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2162
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{2 x^{7/2}}{7 b}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{x^{5/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{2 x^{7/2}}{7 b}+\frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac{x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{2 x^{7/2}}{7 b}+\frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{2 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{3 b^3}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3 \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac{2 x^{7/2}}{7 b}+\frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{2 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{3 b^3}+\frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{b^4}+\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^4 \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{b^4}\\ &=\frac{2 x^{7/2}}{7 b}+\frac{2 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}+\frac{2 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{3 b^3}+\frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}{b^4}-\frac{2 \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 )^{7/2}}{b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0584617, size = 129, normalized size = 0.9 \[ \frac{2 \left (-406 b^{5/2} x^{5/2} \tanh ^{-1}(\tanh (a+b x))+350 b^{3/2} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2-105 \sqrt{b} \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^3+105 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )+176 b^{7/2} x^{7/2}\right )}{105 b^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.132, size = 481, 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 = 2.09325, size = 366, normalized size = 2.56 \begin{align*} \left [\frac{105 \, a^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \, a^{3} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}\right )}}{105 \, b^{4}}\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 = 1.12817, size = 95, normalized size = 0.66 \begin{align*} \frac{2 \, a^{4} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, b^{6} x^{\frac{7}{2}} - 21 \, a b^{5} x^{\frac{5}{2}} + 35 \, a^{2} b^{4} x^{\frac{3}{2}} - 105 \, a^{3} b^{3} \sqrt{x}\right )}}{105 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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