Optimal. Leaf size=151 \[ \frac{x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\sqrt{2} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.139191, antiderivative size = 151, 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, 21, 83, 63, 208, 206} \[ \frac{x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\sqrt{2} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 103
Rule 21
Rule 83
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{3/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{3/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{-\frac{1}{2 a}-\frac{x}{2 a^2}}{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 )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{2 a \left (c-\frac{c}{a x}\right )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \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 )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \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 )^{3/2}}-\frac{\left (2 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \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 )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\sqrt{2} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0680978, size = 91, normalized size = 0.6 \[ \frac{\left (1-\frac{1}{a x}\right )^{3/2} \left (a x \sqrt{\frac{1}{a x}+1}+\tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.18, size = 162, normalized size = 1.1 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{2\,{c}^{2} \left ( ax-1 \right ) }\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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