Optimal. Leaf size=65 \[ 16 b^2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\frac{4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{32}{3} b^3 x^{3/2} \]
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Rubi [A] time = 0.0382775, antiderivative size = 65, 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} \[ 16 b^2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\frac{4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{32}{3} b^3 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^{5/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}+(2 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^{3/2}} \, dx\\ &=-\frac{4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}+\left (8 b^2\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \, dx\\ &=16 b^2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}-\left (16 b^3\right ) \int \sqrt{x} \, dx\\ &=-\frac{32}{3} b^3 x^{3/2}+16 b^2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{4 b \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0294446, size = 55, normalized size = 0.85 \[ -\frac{2 \left (-24 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+6 b x \tanh ^{-1}(\tanh (a+b x))^2+\tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 0.9 \begin{align*} -{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{3}{x}^{-{\frac{3}{2}}}}+4\,b \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{\sqrt{x}}}+4\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \sqrt{x}-2/3\,b{x}^{3/2} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06879, size = 74, normalized size = 1.14 \begin{align*} -\frac{4 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{\sqrt{x}} - \frac{16}{3} \,{\left (2 \, b^{2} x^{\frac{3}{2}} - 3 \, b \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97198, size = 74, normalized size = 1.14 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 9 \, a b^{2} x^{2} - 9 \, a^{2} b x - a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.9452, size = 66, normalized size = 1.02 \begin{align*} - \frac{32 b^{3} x^{\frac{3}{2}}}{3} + 16 b^{2} \sqrt{x} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )} - \frac{4 b \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{\sqrt{x}} - \frac{2 \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16433, size = 46, normalized size = 0.71 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \,{\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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