Optimal. Leaf size=48 \[ -\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{15 x^{3/2}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{5 x^{5/2}}-\frac{16 b^2}{15 \sqrt{x}} \]
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Rubi [A] time = 0.023184, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ -\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{15 x^{3/2}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{5 x^{5/2}}-\frac{16 b^2}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^{7/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{5 x^{5/2}}+\frac{1}{5} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^{5/2}} \, dx\\ &=-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{15 x^{3/2}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{5 x^{5/2}}+\frac{1}{15} \left (8 b^2\right ) \int \frac{1}{x^{3/2}} \, dx\\ &=-\frac{16 b^2}{15 \sqrt{x}}-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{15 x^{3/2}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{5 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0428595, size = 40, normalized size = 0.83 \[ -\frac{2 \left (4 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 38, normalized size = 0.8 \begin{align*} -{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{5}{x}^{-{\frac{5}{2}}}}+{\frac{8\,b}{5} \left ( -{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3}{x}^{-{\frac{3}{2}}}}-{\frac{2\,b}{3}{\frac{1}{\sqrt{x}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05763, size = 49, normalized size = 1.02 \begin{align*} -\frac{16 \, b^{2}}{15 \, \sqrt{x}} - \frac{8 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{15 \, x^{\frac{3}{2}}} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08503, size = 63, normalized size = 1.31 \begin{align*} -\frac{2 \,{\left (15 \, b^{2} x^{2} + 10 \, a b x + 3 \, a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \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 = 1.14518, size = 32, normalized size = 0.67 \begin{align*} -\frac{2 \,{\left (15 \, b^{2} x^{2} + 10 \, a b x + 3 \, a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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