Optimal. Leaf size=113 \[ -\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{8 \sqrt{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+x \left (c-\frac{c}{a x}\right )^{3/2} \]
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Rubi [A] time = 0.238425, antiderivative size = 113, 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, 98, 154, 156, 63, 208} \[ -\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{8 \sqrt{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+x \left (c-\frac{c}{a x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 98
Rule 154
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx\\ &=-\int \frac{\left (c-\frac{c}{a x}\right )^{3/2} (1-a x)}{1+a x} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{5/2} x}{1+a x} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{5/2}}{a+\frac{1}{x}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2}}{x^2 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\left (c-\frac{c}{a x}\right )^{3/2} x+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}} \left (\frac{7 c^2}{2}-\frac{c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+\left (c-\frac{c}{a x}\right )^{3/2} x+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{7 c^3}{4}-\frac{9 c^3 x}{4 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+\left (c-\frac{c}{a x}\right )^{3/2} x+\frac{\left (7 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}-\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+\left (c-\frac{c}{a x}\right )^{3/2} x-(7 c) \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )+(16 c) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+\left (c-\frac{c}{a x}\right )^{3/2} x-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{8 \sqrt{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0829177, size = 95, normalized size = 0.84 \[ \frac{-7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )+8 \sqrt{2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )+c (a x-2) \sqrt{c-\frac{c}{a x}}}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.163, size = 229, normalized size = 2. \begin{align*}{\frac{c}{2\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -10\,\sqrt{a{x}^{2}-x}{a}^{3/2}\sqrt{{a}^{-1}}{x}^{2}+8\,{a}^{3/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}+4\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}\sqrt{{a}^{-1}}+5\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{2}a-8\,\sqrt{a}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){x}^{2}-12\,\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 \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\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{{\left (a x - 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20556, size = 541, normalized size = 4.79 \begin{align*} \left [\frac{8 \, \sqrt{2} c^{\frac{3}{2}} \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 ) + 7 \, c^{\frac{3}{2}} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 2 \,{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{2 \, a}, -\frac{8 \, \sqrt{2} \sqrt{-c} c \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) - 7 \, \sqrt{-c} c \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (a c x - 2 \, c\right )} \sqrt{\frac{a c x - c}{a x}}}{a}\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{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{3}{2}} \left (a x - 1\right )}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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