Optimal. Leaf size=193 \[ \frac{3 a^3 x^3 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{16 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{a^3 x^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}-\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{16 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
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Rubi [A] time = 0.203284, antiderivative size = 193, 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} \[ \frac{3 a^3 x^3 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{16 \left (a-\frac{1}{x}\right ) (c-a c x)^{7/2}}-\frac{a^3 x^2 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}-\frac{3 a^{5/2} \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{16 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]
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^{-\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^{-\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 )^3 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}{4 \left (a-\frac{1}{x}\right )^2 (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 )^2 \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^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{16 \left (a-\frac{1}{x}\right ) (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 )}{32 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{16 \left (a-\frac{1}{x}\right ) (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 )}{16 \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^2}{4 \left (a-\frac{1}{x}\right )^2 (c-a c x)^{7/2}}+\frac{3 a^3 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x^3}{16 \left (a-\frac{1}{x}\right ) (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 )}{16 \sqrt{2} \left (\frac{1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.161128, size = 125, normalized size = 0.65 \[ \frac{x \sqrt{1-\frac{1}{a x}} \left (2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (7-3 a x)+3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{32 \sqrt{a} c^3 (a x-1)^2 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 172, normalized size = 0.9 \begin{align*} -{\frac{ax+1}{32\, \left ( ax-1 \right ) ^{3}a}\sqrt{{\frac{ax-1}{ax+1}}}\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-6\,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+14\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{9}{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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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 = 2.01074, size = 786, normalized size = 4.07 \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 (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{64 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\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 (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{32 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, 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 = 1.22392, size = 123, normalized size = 0.64 \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 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 10 \, \sqrt{-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2} a c^{2}}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{32 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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