Optimal. Leaf size=335 \[ \frac{x \left (a-\frac{1}{x}\right )^5 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{65 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^3 \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{135 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{5 \left (304 a-\frac{65}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{15 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}} \]
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Rubi [A] time = 0.200309, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6182, 6179, 98, 150, 153, 147, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^5 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{65 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^3 \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{135 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{5 \left (304 a-\frac{65}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{15 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 150
Rule 153
Rule 147
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 )^6}{x^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (c-\frac{c}{a x}\right )^{9/2} \operatorname{Subst}\left (\int \frac{\left (\frac{15}{2 a}+\frac{5 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^4}{x \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (a-\frac{1}{x}\right )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (2 a \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{15}{4 a^2}-\frac{65 x}{4 a^3}\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{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{65 \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 )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (4 a^2 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{105}{8 a^3}-\frac{675 x}{8 a^4}\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{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{135 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{65 \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 )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (8 a^3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{525}{16 a^4}-\frac{4875 x}{16 a^5}\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{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{5 \left (304 a-\frac{65}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{135 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{65 \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 )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (15 \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{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{5 \left (304 a-\frac{65}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{135 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{65 \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 )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (15 \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{10 \left (a-\frac{1}{x}\right )^4 \left (c-\frac{c}{a x}\right )^{9/2}}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{5 \left (304 a-\frac{65}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{135 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{65 \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 )^5 \left (c-\frac{c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{15 \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 [C] time = 0.114218, size = 140, normalized size = 0.42 \[ \frac{c^4 \sqrt{c-\frac{c}{a x}} \left (70 a^4 x^4 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{a x}+1\right )+7 a^5 x^5+1685 a^4 x^4+720 a^3 x^3-110 a^2 x^2-35 a^4 x^4 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )+20 a x-2\right )}{7 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 229, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{4}}{14\, \left ( ax-1 \right ) ^{2}{x}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 14\,\sqrt{ \left ( ax+1 \right ) x}{a}^{11/2}{x}^{5}+3510\,{a}^{9/2}\sqrt{ \left ( ax+1 \right ) x}{x}^{4}+1440\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{5}{a}^{5}-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}-220\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+40\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){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{\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.
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Fricas [A] time = 2.21413, size = 918, normalized size = 2.74 \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 (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{28 \,{\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 (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{14 \,{\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{\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.
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