Optimal. Leaf size=184 \[ -\frac{3 a^2 x^3 \left (1-\frac{1}{a x}\right )^{7/2}}{4 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}-\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}+\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{4 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.197943, antiderivative size = 184, 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} \[ -\frac{3 a^2 x^3 \left (1-\frac{1}{a x}\right )^{7/2}}{4 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}-\frac{a^2 x^2 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}+\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{4 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
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)^{7/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{x^{3/2}}{\left (1-\frac{x}{a}\right )^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^2}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}+\frac{\left (3 a \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{4 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^2}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}-\frac{3 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^3}{4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}+\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\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 )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^2}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}-\frac{3 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^3}{4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}+\frac{\left (3 a^2 \left (1-\frac{1}{a x}\right )^{7/2}\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 )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^2}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}-\frac{3 a^2 \left (1-\frac{1}{a x}\right )^{7/2} x^3}{4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{7/2}}+\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \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 )^{7/2} (c-a c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.141503, size = 140, normalized size = 0.76 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (2 \sqrt{\frac{1}{x}} (3 a x-1)-3 \sqrt{2} \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{8 a c^3 \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} (a x-1) \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 129, normalized size = 0.7 \begin{align*} -{\frac{ax+1}{8\, \left ( ax-1 \right ) ^{3}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}xa\sqrt{-c \left ( ax+1 \right ) }-3\,\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}\sqrt{-c \left ( ax+1 \right ) }+6\,xa\sqrt{c}-2\,\sqrt{c} \right ){c}^{-{\frac{9}{2}}}} \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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (-a c x + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58982, size = 675, normalized size = 3.67 \begin{align*} \left [-\frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 \, \sqrt{-a c x + c}{\left (3 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{16 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, 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 \, \sqrt{-a c x + c}{\left (3 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\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 = 1.21681, size = 122, normalized size = 0.66 \begin{align*} -\frac{{\left (\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{a c^{\frac{5}{2}}} - \frac{2 \,{\left (3 \, a c x - c\right )}}{{\left ({\left (-a c x - c\right )}^{\frac{3}{2}} + 2 \, \sqrt{-a c x - c} c\right )} a c^{2}}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{8 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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