Optimal. Leaf size=69 \[ \frac{16}{429} b^2 x^{13/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{33} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{15/2}}{6435} \]
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Rubi [A] time = 0.0385126, 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} \[ \frac{16}{429} b^2 x^{13/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{33} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{32 b^3 x^{15/2}}{6435} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{7/2} \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{3} (2 b) \int x^{9/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{4}{33} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{33} \left (8 b^2\right ) \int x^{11/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{16}{429} b^2 x^{13/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{33} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{429} \left (16 b^3\right ) \int x^{13/2} \, dx\\ &=-\frac{32 b^3 x^{15/2}}{6435}+\frac{16}{429} b^2 x^{13/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{33} b x^{11/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{2}{9} x^{9/2} \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.0289775, size = 57, normalized size = 0.83 \[ -\frac{2 x^{9/2} \left (-120 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+390 b x \tanh ^{-1}(\tanh (a+b x))^2-715 \tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{6435} \]
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}}{9}{x}^{{\frac{9}{2}}}}-{\frac{4\,b}{3} \left ({\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{11}{x}^{{\frac{11}{2}}}}-{\frac{4\,b}{11} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{13}{x}^{{\frac{13}{2}}}}-{\frac{2\,b}{195}{x}^{{\frac{15}{2}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06234, size = 74, normalized size = 1.07 \begin{align*} -\frac{4}{33} \, b x^{\frac{11}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{2}{9} \, x^{\frac{9}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{16}{6435} \,{\left (2 \, b^{2} x^{\frac{15}{2}} - 15 \, b x^{\frac{13}{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 = 1.92607, size = 105, normalized size = 1.52 \begin{align*} \frac{2}{6435} \,{\left (429 \, b^{3} x^{7} + 1485 \, a b^{2} x^{6} + 1755 \, a^{2} b x^{5} + 715 \, a^{3} x^{4}\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 = 1.14243, size = 47, normalized size = 0.68 \begin{align*} \frac{2}{15} \, b^{3} x^{\frac{15}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{2}{9} \, a^{3} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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