Optimal. Leaf size=172 \[ \frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a c^{9/2}} \]
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Rubi [A] time = 0.293287, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 103, 152, 156, 63, 208} \[ \frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a c^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 103
Rule 152
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{9/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{9/2}} \, dx\\ &=-\int \frac{1-a x}{\left (c-\frac{c}{a x}\right )^{9/2} (1+a x)} \, dx\\ &=\frac{a \int \frac{x}{\left (c-\frac{c}{a x}\right )^{7/2} (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{1}{\left (a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x) \left (c-\frac{c x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{5 c}{2}-\frac{7 c x}{2 a}}{x (a+x) \left (c-\frac{c x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{25 c^2}{2}+\frac{15 c^2 x}{a}}{x (a+x) \left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^4}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{75 c^3}{2}-\frac{165 c^3 x}{4 a}}{x (a+x) \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^6}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{75 c^4}{2}+\frac{315 c^4 x}{8 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{15 c^8}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a c^4}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c^4}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{4 c^5}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^5}\\ &=-\frac{6}{5 a c^2 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{6 a c^3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{9/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a c^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0649847, size = 82, normalized size = 0.48 \[ \frac{a x^2 \left (-\text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},\frac{a-\frac{1}{x}}{2 a}\right )-5 \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},1-\frac{1}{a x}\right )+5 a x\right )}{5 c^4 (a x-1)^2 \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.174, size = 626, normalized size = 3.6 \begin{align*} \text{result too large to display} \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{a x - 1}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00998, size = 976, normalized size = 5.67 \begin{align*} \left [\frac{15 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 600 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, 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 ) + 4 \,{\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{240 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, -\frac{15 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) + 600 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) - 2 \,{\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{120 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\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 [A] time = 1.34761, size = 279, normalized size = 1.62 \begin{align*} -\frac{1}{120} \, a c{\left (\frac{2 \,{\left (12 \, c^{2} + \frac{50 \,{\left (a c x - c\right )} c}{a x} + \frac{255 \,{\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c^{5} \sqrt{\frac{a c x - c}{a x}}} + \frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a c x - c}{a x}}}{2 \, \sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{5}} + \frac{600 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{5}} - \frac{120 \, \sqrt{\frac{a c x - c}{a x}}}{a^{2}{\left (c - \frac{a c x - c}{a x}\right )} c^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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