Optimal. Leaf size=160 \[ -\frac{15 \sqrt{c-a c x}}{32 a c^5 \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}+\frac{1}{4 a c^3 \sqrt{1-a^2 x^2} (c-a c x)^{3/2}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{9/2}} \]
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Rubi [A] time = 0.129878, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6127, 673, 667, 661, 208} \[ -\frac{15 \sqrt{c-a c x}}{32 a c^5 \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{1-a^2 x^2} \sqrt{c-a c x}}+\frac{1}{4 a c^3 \sqrt{1-a^2 x^2} (c-a c x)^{3/2}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 673
Rule 667
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx &=\frac{\int \frac{1}{(c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{1}{4 a c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}+\frac{5 \int \frac{1}{\sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}} \, dx}{8 c^4}\\ &=\frac{1}{4 a c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}+\frac{15 \int \frac{\sqrt{c-a c x}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{32 c^5}\\ &=\frac{1}{4 a c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{c-a c x}}{32 a c^5 \sqrt{1-a^2 x^2}}+\frac{15 \int \frac{1}{\sqrt{c-a c x} \sqrt{1-a^2 x^2}} \, dx}{64 c^4}\\ &=\frac{1}{4 a c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{c-a c x}}{32 a c^5 \sqrt{1-a^2 x^2}}-\frac{(15 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 )}{32 c^3}\\ &=\frac{1}{4 a c^3 (c-a c x)^{3/2} \sqrt{1-a^2 x^2}}+\frac{5}{16 a c^4 \sqrt{c-a c x} \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{c-a c x}}{32 a c^5 \sqrt{1-a^2 x^2}}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{2} \sqrt{c-a c x}}\right )}{32 \sqrt{2} a c^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0299716, size = 57, normalized size = 0.36 \[ -\frac{(1-a x)^{3/2} \text{Hypergeometric2F1}\left (-\frac{1}{2},3,\frac{1}{2},\frac{1}{2} (a x+1)\right )}{4 a c^3 \sqrt{a x+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.119, size = 173, normalized size = 1.1 \begin{align*} -{\frac{1}{64\, \left ( ax-1 \right ) ^{3} \left ( ax+1 \right ) a}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }-30\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) xa\sqrt{c \left ( ax+1 \right ) }-30\,{x}^{2}{a}^{2}\sqrt{c}+15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}\sqrt{c \left ( ax+1 \right ) }+40\,xa\sqrt{c}+6\,\sqrt{c} \right ){c}^{-{\frac{11}{2}}}} \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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (-a c x + c\right )}^{\frac{9}{2}}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63633, size = 747, normalized size = 4.67 \begin{align*} \left [\frac{15 \, \sqrt{2}{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, 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 \,{\left (15 \, a^{2} x^{2} - 20 \, a x - 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{128 \,{\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right )}}, \frac{15 \, \sqrt{2}{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, 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 \,{\left (15 \, a^{2} x^{2} - 20 \, a x - 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{64 \,{\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\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 [A] time = 1.30077, size = 134, normalized size = 0.84 \begin{align*} -\frac{{\left (\frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c} c^{3}} + \frac{16}{\sqrt{a c x + c} a c^{3}} + \frac{2 \,{\left (7 \,{\left (a c x + c\right )}^{\frac{3}{2}} - 18 \, \sqrt{a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} a c^{3}}\right )}{\left | c \right |}}{64 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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