Optimal. Leaf size=235 \[ \frac{173 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}{105 a \sqrt{c-\frac{c}{a x}}}+\frac{227 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{105 a}+\frac{59 c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 a}+\frac{9 c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 a}-\frac{7 c^{9/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}+c x \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{7/2} \]
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Rubi [A] time = 0.163788, antiderivative size = 279, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6182, 6179, 97, 153, 147, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^4 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{59 \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (400 a-\frac{227}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{7 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6182
Rule 6179
Rule 97
Rule 153
Rule 147
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\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^{\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 )^4 \sqrt{1+\frac{x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \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{7}{2 a}-\frac{9 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^3}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \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{49}{4 a^2}-\frac{59 x}{4 a^3}\right ) \left (1-\frac{x}{a}\right )^2}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{7 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{59 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (4 a^2 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{245}{8 a^3}-\frac{227 x}{8 a^4}\right ) \left (1-\frac{x}{a}\right )}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{35 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{\left (400 a-\frac{227}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{59 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (7 \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{\left (400 a-\frac{227}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{59 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (7 \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{\left (400 a-\frac{227}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{105 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{59 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^4 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{7 \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 = 0.114718, size = 109, normalized size = 0.46 \[ \frac{c^4 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (105 a^4 x^4+292 a^3 x^3-356 a^2 x^2+162 a x-30\right )-735 a^3 x^3 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{105 a^4 x^3 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.178, size = 166, normalized size = 0.7 \begin{align*} -{\frac{{c}^{4}}{210\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -210\,{a}^{9/2}\sqrt{ \left ( ax+1 \right ) x}{x}^{4}+735\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{4}{a}^{4}-584\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+712\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-324\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+60\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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 (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19396, size = 926, normalized size = 3.94 \begin{align*} \left [\frac{735 \,{\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 (105 \, a^{5} c^{4} x^{5} + 397 \, a^{4} c^{4} x^{4} - 64 \, a^{3} c^{4} x^{3} - 194 \, a^{2} c^{4} x^{2} + 132 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{420 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac{735 \,{\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 (105 \, a^{5} c^{4} x^{5} + 397 \, a^{4} c^{4} x^{4} - 64 \, a^{3} c^{4} x^{3} - 194 \, a^{2} c^{4} x^{2} + 132 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{210 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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