Optimal. Leaf size=127 \[ \frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3} \]
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Rubi [A] time = 0.277919, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{\sqrt{c-a c x}}{3 x^3} \]
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^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{x^4 (1+a x)} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x^4 (1+a x)} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}+\frac{\int \frac{\frac{13 a c^2}{2}-\frac{11}{2} a^2 c^2 x}{x^3 (1+a x) \sqrt{c-a c x}} \, dx}{3 c}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}-\frac{\int \frac{\frac{57 a^2 c^3}{4}-\frac{39}{4} a^3 c^3 x}{x^2 (1+a x) \sqrt{c-a c x}} \, dx}{6 c^2}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}+\frac{\int \frac{\frac{135 a^3 c^4}{8}-\frac{57}{8} a^4 c^4 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{6 c^3}\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}+\frac{1}{16} \left (45 a^3 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx-\left (4 a^4 c\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{1}{8} \left (45 a^2\right ) \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^3\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}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}+\frac{19 a^2 \sqrt{c-a c x}}{8 x}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0902037, size = 101, normalized size = 0.8 \[ \frac{\left (57 a^2 x^2-26 a x+8\right ) \sqrt{c-a c x}}{24 x^3}-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 110, normalized size = 0.9 \begin{align*} -2\,{c}^{3}{a}^{3} \left ( -{\frac{1}{{c}^{2}} \left ( -{\frac{1}{{x}^{3}{a}^{3}{c}^{3}} \left ( -{\frac{19\, \left ( -acx+c \right ) ^{5/2}}{16}}+{\frac{11\,c \left ( -acx+c \right ) ^{3/2}}{6}}-{\frac{13\,\sqrt{-acx+c}{c}^{2}}{16}} \right ) }-{\frac{45}{16\,\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) }-2\,{\frac{\sqrt{2}}{{c}^{5/2}}{\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 = 1.37009, size = 568, normalized size = 4.47 \begin{align*} \left [\frac{96 \, \sqrt{2} a^{3} \sqrt{c} x^{3} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 135 \, a^{3} \sqrt{c} x^{3} \log \left (\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \,{\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt{-a c x + c}}{48 \, x^{3}}, -\frac{96 \, \sqrt{2} a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 135 \, a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt{-a c x + c}}{24 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 16.9645, size = 614, normalized size = 4.83 \begin{align*} - \frac{66 a^{3} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac{80 a^{3} c^{5} \left (- a c x + c\right )^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{30 a^{3} c^{4} \left (- a c x + c\right )^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{30 a^{3} 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{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{18 a^{3} 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{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - \frac{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )} + 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )} + \frac{8 a^{3} c \operatorname{atan}{\left (\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right )}}{\sqrt{- c}} - \frac{4 \sqrt{2} a^{3} c \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + \frac{4 a^{2} \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 = 1.13983, size = 180, normalized size = 1.42 \begin{align*} -\frac{4 \, \sqrt{2} a^{3} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + \frac{45 \, a^{3} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c}} + \frac{57 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{3} c - 88 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{3} c^{2} + 39 \, \sqrt{-a c x + c} a^{3} c^{3}}{24 \, a^{3} c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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