Optimal. Leaf size=69 \[ \frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{13/2}}{3003} \]
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Rubi [A] time = 0.0377565, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ \frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{13/2}}{3003} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{7} (6 b) \int x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{21} \left (8 b^2\right ) \int x^{9/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{231} \left (16 b^3\right ) \int x^{11/2} \, dx\\ &=-\frac{32 b^3 x^{13/2}}{3003}+\frac{16}{231} b^2 x^{11/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{21} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.0393816, size = 57, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (104 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))-286 b x \tanh ^{-1}(\tanh (a+b x))^2+429 \tanh ^{-1}(\tanh (a+b x))^3-16 b^3 x^3\right )}{3003} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 56, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{7}{x}^{{\frac{7}{2}}}}-{\frac{12\,b}{7} \left ({\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{9}{x}^{{\frac{9}{2}}}}-{\frac{4\,b}{9} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{11}{x}^{{\frac{11}{2}}}}-{\frac{2\,b}{143}{x}^{{\frac{13}{2}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06479, size = 74, normalized size = 1.07 \begin{align*} -\frac{4}{21} \, b x^{\frac{9}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{2}{7} \, x^{\frac{7}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{16}{3003} \,{\left (2 \, b^{2} x^{\frac{13}{2}} - 13 \, b x^{\frac{11}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04076, size = 104, normalized size = 1.51 \begin{align*} \frac{2}{3003} \,{\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt{x} \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 = 1.13728, size = 47, normalized size = 0.68 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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