Optimal. Leaf size=268 \[ \frac{9 \left (a-\frac{1}{x}\right )^2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.163274, antiderivative size = 268, 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 = {6182, 6179, 97, 153, 147, 50, 63, 208} \[ \frac{9 \left (a-\frac{1}{x}\right )^2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6182
Rule 6179
Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{9/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{9/2} \, dx}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{9/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3 \left (1+\frac{x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (c-\frac{c}{a x}\right )^{9/2} \operatorname{Subst}\left (\int \frac{\left (-\frac{3}{2 a}-\frac{9 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (2 a \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{21}{4 a^2}-\frac{51 x}{4 a^3}\right ) \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{7 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.127988, size = 109, normalized size = 0.41 \[ \frac{c^4 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (35 a^4 x^4+164 a^3 x^3-12 a^2 x^2-26 a x+10\right )-105 a^3 x^3 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{35 a^4 x^3 \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.181, size = 178, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{4}}{ \left ( 70\,ax+70 \right ){x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -70\,{a}^{9/2}\sqrt{ \left ( ax+1 \right ) x}{x}^{4}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{4}{a}^{4}-328\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+24\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+52\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-20\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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{{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.32026, size = 919, normalized size = 3.43 \begin{align*} \left [\frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{140 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{70 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]