Optimal. Leaf size=275 \[ \frac{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{9 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{51 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.171335, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6182, 6179, 98, 151, 156, 63, 208, 206} \[ \frac{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{9 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{51 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 151
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 )^{3/2}} \, dx &=\frac{\left (1-\frac{1}{a x}\right )^{3/2} \int \frac{e^{3 \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{\left (1+\frac{x}{a}\right )^{3/2}}{x^2 \left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{3/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^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{-\frac{9}{2 a}-\frac{7 x}{2 a^2}}{x \left (1-\frac{x}{a}\right )^3 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{3/2}}\\ &=-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (a \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{18}{a^2}+\frac{12 x}{a^3}}{x \left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 \left (c-\frac{c}{a x}\right )^{3/2}}\\ &=-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{\left (a^2 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{36}{a^3}-\frac{15 x}{a^4}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \left (c-\frac{c}{a x}\right )^{3/2}}\\ &=-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (51 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (9 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \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{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{\left (9 \left (1-\frac{1}{a x}\right )^{3/2}\right ) \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 (51 \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 )}{4 a \left (c-\frac{c}{a x}\right )^{3/2}}\\ &=-\frac{2 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{15 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{9 \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{51 \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.12598, size = 135, normalized size = 0.49 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (2 a x \sqrt{\frac{1}{a x}+1} \left (4 a^2 x^2-23 a x+15\right )+72 (a x-1)^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-51 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{8 a c (a x-1)^2 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.186, size = 373, normalized size = 1.4 \begin{align*}{\frac{x}{ \left ( 16\,ax-16 \right ) \left ( ax+1 \right ){c}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}{x}^{2}-51\,{a}^{5/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ){x}^{2}-92\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+72\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{3}\sqrt{{a}^{-1}}{x}^{2}+102\,{a}^{3/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) x-144\,\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+60\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+72\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}-51\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\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{1}{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{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 = 2.69249, size = 1455, normalized size = 5.29 \begin{align*} \left [\frac{51 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 72 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\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 ) + 8 \,{\left (4 \, a^{4} x^{4} - 19 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 15 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{32 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac{51 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{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}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 72 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\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 ) + 4 \,{\left (4 \, a^{4} x^{4} - 19 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 15 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\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{1}{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{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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