Optimal. Leaf size=106 \[ \frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )+\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x} \]
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Rubi [A] time = 0.259572, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {6167, 6130, 21, 98, 151, 156, 63, 208, 206} \[ \frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )+\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{x^3 (1+a x)} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x^3 (1+a x)} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{2 x^2}+\frac{\int \frac{\frac{9 a c^2}{2}-\frac{7}{2} a^2 c^2 x}{x^2 (1+a x) \sqrt{c-a c x}} \, dx}{2 c}\\ &=\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x}-\frac{\int \frac{\frac{23 a^2 c^3}{4}-\frac{9}{4} a^3 c^3 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{2 c^2}\\ &=\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x}-\frac{1}{8} \left (23 a^2 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx+\left (4 a^3 c\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x}+\frac{1}{4} (23 a) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )-\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x}+\frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0766494, size = 93, normalized size = 0.88 \[ \frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )+\frac{(2-9 a x) \sqrt{c-a c x}}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 95, normalized size = 0.9 \begin{align*} 2\,{a}^{2}{c}^{2} \left ( -{\frac{1}{c} \left ({\frac{1}{{a}^{2}{x}^{2}{c}^{2}} \left ( -{\frac{9\, \left ( -acx+c \right ) ^{3/2}}{8}}+{\frac{7\,c\sqrt{-acx+c}}{8}} \right ) }-{\frac{23}{8\,\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) }-2\,{\frac{\sqrt{2}}{{c}^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25115, size = 524, normalized size = 4.94 \begin{align*} \left [\frac{16 \, \sqrt{2} a^{2} \sqrt{c} x^{2} \log \left (\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 23 \, a^{2} \sqrt{c} x^{2} \log \left (\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) - 2 \, \sqrt{-a c x + c}{\left (9 \, a x - 2\right )}}{8 \, x^{2}}, \frac{16 \, \sqrt{2} a^{2} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 23 \, a^{2} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) - \sqrt{-a c x + c}{\left (9 \, a x - 2\right )}}{4 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.8379, size = 352, normalized size = 3.32 \begin{align*} \frac{10 a^{2} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac{6 a^{2} c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} + \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} + \frac{3 a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} - \frac{3 a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} - \frac{8 a^{2} c \operatorname{atan}{\left (\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + \frac{4 \sqrt{2} a^{2} c \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} - \frac{3 a \sqrt{- a c x + c}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14123, size = 143, normalized size = 1.35 \begin{align*} \frac{4 \, \sqrt{2} a^{2} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} - \frac{23 \, a^{2} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} + \frac{9 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{2} c - 7 \, \sqrt{-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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