Optimal. Leaf size=70 \[ \frac{x}{\sqrt{c-\frac{c}{a x}}}-\frac{5}{a \sqrt{c-\frac{c}{a x}}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.179245, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 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.
[In]
[Out]
Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=-\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 = 0.0303269, size = 43, normalized size = 0.61 \[ \frac{a x-5 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},1-\frac{1}{a x}\right )}{a \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.177, size = 194, normalized size = 2.8 \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.91305, size = 377, normalized size = 5.39 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.30974, size = 167, normalized size = 2.39 \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.
[In]
[Out]