Optimal. Leaf size=71 \[ -\frac{x}{\sqrt{c-\frac{c}{a x}}}+\frac{5}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.162234, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 78, 51, 63, 208} \[ -\frac{x}{\sqrt{c-\frac{c}{a x}}}+\frac{5}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\int \frac{1+a x}{\sqrt{c-\frac{c}{a x}} (1-a x)} \, dx\\ &=-\frac{c \int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{3/2} x} \, dx}{a}\\ &=-\frac{c \int \frac{a+\frac{1}{x}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^2 \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{x}{\sqrt{c-\frac{c}{a x}}}+\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{5}{a \sqrt{c-\frac{c}{a x}}}-\frac{x}{\sqrt{c-\frac{c}{a x}}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{5}{a \sqrt{c-\frac{c}{a x}}}-\frac{x}{\sqrt{c-\frac{c}{a x}}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c}\\ &=\frac{5}{a \sqrt{c-\frac{c}{a x}}}-\frac{x}{\sqrt{c-\frac{c}{a x}}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [C] time = 0.0263909, size = 44, normalized size = 0.62 \[ \frac{5 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},1-\frac{1}{a x}\right )-a x}{a \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.144, size = 194, normalized size = 2.7 \begin{align*} -{\frac{x}{2\,c \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 10\,{a}^{5/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}+5\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}-8\,{a}^{3/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}-20\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}x-10\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) xa+10\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+5\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \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 )}^{2}}{{\left (a^{2} x^{2} - 1\right )} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10614, size = 375, normalized size = 5.28 \begin{align*} \left [\frac{5 \,{\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 ) - 2 \,{\left (a^{2} x^{2} - 5 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} c x - a c\right )}}, \frac{5 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (a^{2} x^{2} - 5 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{a^{2} c x - a c}\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}{a x \sqrt{c - \frac{c}{a x}} - \sqrt{c - \frac{c}{a x}}}\, dx - \int \frac{1}{a x \sqrt{c - \frac{c}{a x}} - \sqrt{c - \frac{c}{a x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25152, size = 166, normalized size = 2.34 \begin{align*} a c{\left (\frac{5 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c} + \frac{4 \, c - \frac{5 \,{\left (a c x - c\right )}}{a x}}{{\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}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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