Optimal. Leaf size=75 \[ -\frac{2 x}{5 c^2 \sqrt{c-a^2 c x^2}}-\frac{x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.11495, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6167, 6142, 653, 192, 191} \[ -\frac{2 x}{5 c^2 \sqrt{c-a^2 c x^2}}-\frac{x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6142
Rule 653
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\right )\\ &=\frac{2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac{3}{5} \int \frac{1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac{x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac{2 (1-a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}-\frac{x}{5 c \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 x}{5 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0534234, size = 79, normalized size = 1.05 \[ -\frac{\sqrt{1-a^2 x^2} \left (2 a^3 x^3+4 a^2 x^2+a x-2\right )}{5 a c^2 \sqrt{1-a x} (a x+1)^{5/2} \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 47, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2} \left ( 2\,{x}^{3}{a}^{3}+4\,{a}^{2}{x}^{2}+ax-2 \right ) }{5\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \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 = 1.89237, size = 151, normalized size = 2.01 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 2\right )} \sqrt{-a^{2} c x^{2} + c}}{5 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\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{a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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