Optimal. Leaf size=116 \[ \frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}} \]
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Rubi [A] time = 0.23544, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 103, 152, 156, 63, 208} \[ \frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 103
Rule 152
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx\\ &=-\int \frac{1-a x}{\left (c-\frac{c}{a x}\right )^{5/2} (1+a x)} \, dx\\ &=\frac{a \int \frac{x}{\left (c-\frac{c}{a x}\right )^{3/2} (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{1}{\left (a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x) \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{c}{2}-\frac{3 c x}{2 a}}{x (a+x) \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{c^2}{2}+\frac{c^2 x}{a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c^2}\\ &=-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^3}\\ &=-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0547699, size = 70, normalized size = 0.6 \[ \frac{-\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{a-\frac{1}{x}}{2 a}\right )-\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},1-\frac{1}{a x}\right )+a x}{a c^2 \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.17, size = 368, normalized size = 3.2 \begin{align*} -{\frac{x}{4\,{c}^{3} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -8\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}+\ln \left ({\frac{1}{ax+1} \left ( 2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1 \right ) } \right ){a}^{{\frac{5}{2}}}\sqrt{2}{x}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{2}{a}^{3}+4\,{a}^{5/2}\sqrt{{a}^{-1}} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}+16\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}x-2\,\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){a}^{3/2}\sqrt{2}x+4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}x{a}^{2}-8\,\sqrt{ \left ( ax-1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+\sqrt{2}\ln \left ({\frac{1}{ax+1} \left ( 2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1 \right ) } \right ) \sqrt{a}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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{a x - 1}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00711, size = 652, normalized size = 5.62 \begin{align*} \left [\frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 2 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 4 \,{\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{2} c^{3} x - a c^{3}\right )}}, -\frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) + 2 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) - 2 \,{\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{5}{2}} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25974, size = 224, normalized size = 1.93 \begin{align*} -\frac{1}{2} \, a c{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a c x - c}{a x}}}{2 \, \sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{3}} + \frac{2 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{3}} + \frac{2 \,{\left (c - \frac{2 \,{\left (a c x - c\right )}}{a x}\right )}}{{\left (c \sqrt{\frac{a c x - c}{a x}} - \frac{{\left (a c x - c\right )} \sqrt{\frac{a c x - c}{a x}}}{a x}\right )} a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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