Optimal. Leaf size=187 \[ -\frac{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{3 a x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{3 \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{4 \sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.193826, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{3 a x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{3 \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{4 \sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{\sqrt{x} \left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{\left (3 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{4 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{3 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{4 \left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{\left (3 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{3 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{4 \left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{\left (3 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{4 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{3 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{4 \left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x}{2 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{3/2}}-\frac{3 \sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{4 \sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.14897, size = 125, normalized size = 0.67 \[ \frac{x \sqrt{1-\frac{1}{a x}} \left (2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (5 a x-1)+3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{8 \sqrt{a} c (a x-1)^2 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.213, size = 174, normalized size = 0.9 \begin{align*} -{\frac{1}{ \left ( 8\,ax-8 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c-6\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac+10\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+3\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c-2\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a c x + c\right )}^{\frac{3}{2}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65276, size = 784, normalized size = 4.19 \begin{align*} \left [-\frac{3 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \,{\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{16 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}, -\frac{3 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) + 2 \,{\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \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 = 1.20051, size = 120, normalized size = 0.64 \begin{align*} -\frac{\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{\sqrt{c}} - \frac{2 \,{\left (5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 6 \, \sqrt{-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2}}}{8 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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