Optimal. Leaf size=83 \[ -\frac{1}{2 a c^3 \sqrt{c-a c x}}-\frac{1}{3 a c^2 (c-a c x)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{2 \sqrt{2} a c^{7/2}} \]
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Rubi [A] time = 0.115963, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6130, 21, 51, 63, 206} \[ -\frac{1}{2 a c^3 \sqrt{c-a c x}}-\frac{1}{3 a c^2 (c-a c x)^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{2 \sqrt{2} a c^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\\ &=-\int \frac{1-a x}{(1+a x) (c-a c x)^{7/2}} \, dx\\ &=-\frac{\int \frac{1}{(1+a x) (c-a c x)^{5/2}} \, dx}{c}\\ &=-\frac{1}{3 a c^2 (c-a c x)^{3/2}}-\frac{\int \frac{1}{(1+a x) (c-a c x)^{3/2}} \, dx}{2 c^2}\\ &=-\frac{1}{3 a c^2 (c-a c x)^{3/2}}-\frac{1}{2 a c^3 \sqrt{c-a c x}}-\frac{\int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{4 c^3}\\ &=-\frac{1}{3 a c^2 (c-a c x)^{3/2}}-\frac{1}{2 a c^3 \sqrt{c-a c x}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{2 a c^4}\\ &=-\frac{1}{3 a c^2 (c-a c x)^{3/2}}-\frac{1}{2 a c^3 \sqrt{c-a c x}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{2 \sqrt{2} a c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0276531, size = 39, normalized size = 0.47 \[ -\frac{\text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\frac{1}{2} (1-a x)\right )}{3 a c^2 (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 64, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{ac} \left ( 1/4\,{\frac{1}{\sqrt{-acx+c}{c}^{2}}}+1/6\,{\frac{1}{c \left ( -acx+c \right ) ^{3/2}}}-1/8\,{\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 = 1.90501, size = 478, normalized size = 5.76 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 4 \, \sqrt{-a c x + c}{\left (3 \, a x - 5\right )}}{24 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 2 \, \sqrt{-a c x + c}{\left (3 \, a x - 5\right )}}{12 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 48.9461, size = 82, normalized size = 0.99 \begin{align*} - \frac{1}{3 a c^{2} \left (- a c x + c\right )^{\frac{3}{2}}} - \frac{1}{2 a c^{3} \sqrt{- a c x + c}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{4 a c^{3} \sqrt{- c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12433, size = 99, normalized size = 1.19 \begin{align*} -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{4 \, a \sqrt{-c} c^{3}} - \frac{3 \, a c x - 5 \, c}{6 \,{\left (a c x - c\right )} \sqrt{-a c x + c} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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