Optimal. Leaf size=69 \[ \frac{16}{35} b^2 x^{7/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{5} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32}{315} b^3 x^{9/2} \]
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Rubi [A] time = 0.0369685, 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} \[ \frac{16}{35} b^2 x^{7/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{5} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32}{315} b^3 x^{9/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3-(2 b) \int x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{4}{5} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{5} \left (8 b^2\right ) \int x^{5/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{16}{35} b^2 x^{7/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{5} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{35} \left (16 b^3\right ) \int x^{7/2} \, dx\\ &=-\frac{32}{315} b^3 x^{9/2}+\frac{16}{35} b^2 x^{7/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{5} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.0276033, size = 57, normalized size = 0.83 \[ -\frac{2}{315} x^{3/2} \left (-72 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+126 b x \tanh ^{-1}(\tanh (a+b x))^2-105 \tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right ) \]
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}}{3}{x}^{{\frac{3}{2}}}}-4\,b \left ( 1/5\,{x}^{5/2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}-4/5\,b \left ( 1/7\,{x}^{7/2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -{\frac{2\,b{x}^{9/2}}{63}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05312, size = 74, normalized size = 1.07 \begin{align*} -\frac{4}{5} \, b x^{\frac{5}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{16}{315} \,{\left (2 \, b^{2} x^{\frac{9}{2}} - 9 \, b x^{\frac{7}{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 = 2.03469, size = 97, normalized size = 1.41 \begin{align*} \frac{2}{315} \,{\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\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 \sqrt{x} \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 = 1.16223, size = 47, normalized size = 0.68 \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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