Optimal. Leaf size=125 \[ \frac{3 \sqrt{1-a^2 x^2}}{16 a c^2 (c-a c x)^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}} \]
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Rubi [A] time = 0.104941, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6127, 673, 661, 208} \[ \frac{3 \sqrt{1-a^2 x^2}}{16 a c^2 (c-a c x)^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{7/2}}+\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 673
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac{\int \frac{1}{(c-a c x)^{5/2} \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}}+\frac{3 \int \frac{1}{(c-a c x)^{3/2} \sqrt{1-a^2 x^2}} \, dx}{8 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{16 a c^2 (c-a c x)^{3/2}}+\frac{3 \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{32 c^3}\\ &=\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{16 a c^2 (c-a c x)^{3/2}}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )}{16 c^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{4 a c (c-a c x)^{5/2}}+\frac{3 \sqrt{1-a^2 x^2}}{16 a c^2 (c-a c x)^{3/2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{16 \sqrt{2} a c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0237634, size = 52, normalized size = 0.42 \[ \frac{\sqrt{1-a^2 x^2} \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},\frac{1}{2} (a x+1)\right )}{4 a c^3 \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.106, size = 158, normalized size = 1.3 \begin{align*} -{\frac{1}{32\, \left ( ax-1 \right ) ^{3}a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c-6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xac-6\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c+14\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \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{-a^{2} x^{2} + 1}}{{\left (-a c x + c\right )}^{\frac{7}{2}}{\left (a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20508, size = 705, normalized size = 5.64 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (3 \, a x - 7\right )}}{64 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac{3 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (3 \, a x - 7\right )}}{32 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (a x - 1\right )\right )^{\frac{7}{2}} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46088, size = 112, normalized size = 0.9 \begin{align*} -\frac{{\left (\frac{3 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c} c^{2}} + \frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{\frac{3}{2}} - 10 \, \sqrt{a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} a c^{2}}\right )}{\left | c \right |}}{32 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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