Optimal. Leaf size=69 \[ \frac{16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac{12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{11/2}}{1155} \]
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Rubi [A] time = 0.0374426, 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} \[ \frac{16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac{12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{11/2}}{1155} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{5} (6 b) \int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{35} \left (24 b^2\right ) \int x^{7/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac{12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{105} \left (16 b^3\right ) \int x^{9/2} \, dx\\ &=-\frac{32 b^3 x^{11/2}}{1155}+\frac{16}{105} b^2 x^{9/2} \tanh ^{-1}(\tanh (a+b x))-\frac{12}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.0316131, size = 57, normalized size = 0.83 \[ -\frac{2 x^{5/2} \left (-88 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+198 b x \tanh ^{-1}(\tanh (a+b x))^2-231 \tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 56, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{5}{x}^{{\frac{5}{2}}}}-{\frac{12\,b}{5} \left ({\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{7}{x}^{{\frac{7}{2}}}}-{\frac{4\,b}{7} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{9}{x}^{{\frac{9}{2}}}}-{\frac{2\,b}{99}{x}^{{\frac{11}{2}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06569, size = 74, normalized size = 1.07 \begin{align*} -\frac{12}{35} \, b x^{\frac{7}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{2}{5} \, x^{\frac{5}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{16}{1155} \,{\left (2 \, b^{2} x^{\frac{11}{2}} - 11 \, b x^{\frac{9}{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 = 2.02683, size = 103, normalized size = 1.49 \begin{align*} \frac{2}{1155} \,{\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} \sqrt{x} \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 x^{\frac{3}{2}} \operatorname{atanh}^{3}{\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 = 1.17472, size = 47, normalized size = 0.68 \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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