Optimal. Leaf size=199 \[ \frac{x \left (1-\frac{1}{a x}\right )^{5/2}}{\sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{2} a \left (c-\frac{c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.167721, antiderivative size = 199, 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, 103, 152, 156, 63, 208, 206} \[ \frac{x \left (1-\frac{1}{a x}\right )^{5/2}}{\sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{2} a \left (c-\frac{c}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 103
Rule 152
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx &=\frac{\left (1-\frac{1}{a x}\right )^{5/2} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{5/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{5/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{5/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{5/2} x}{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{\frac{1}{2 a}-\frac{3 x}{2 a^2}}{x \left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{5/2}}\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} x}{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (a \left (1-\frac{1}{a x}\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2 a^2}-\frac{x}{a^3}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{5/2}}\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} x}{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a^2 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (c-\frac{c}{a x}\right )^{5/2}}\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} x}{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a \left (c-\frac{c}{a x}\right )^{5/2}}\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{5/2}}{a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{\left (1-\frac{1}{a x}\right )^{5/2} x}{\sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{\left (1-\frac{1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{\sqrt{2} a \left (c-\frac{c}{a x}\right )^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0624847, size = 90, normalized size = 0.45 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{a+\frac{1}{x}}{2 a}\right )+\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{a x}+1\right )+a x\right )}{a c^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.187, size = 262, normalized size = 1.3 \begin{align*} -{\frac{ \left ( ax+1 \right ) x}{4\, \left ( ax-1 \right ) ^{2}{c}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -4\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{2}\sqrt{{a}^{-1}}x+{a}^{{\frac{3}{2}}}\sqrt{2}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1 \right ) } \right ) x-8\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}+\sqrt{2}\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1 \right ) } \right ) \sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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 (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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