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 = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 191
Rule 192
Rule 653
Rule 6142
Rule 6167
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.57, size = 75, normalized size = 1.00 \[ \frac {{\left (2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x - 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 47, normalized size = 0.63 \[ -\frac {\left (a x -1\right )^{2} \left (2 x^{3} a^{3}+4 a^{2} x^{2}+a x -2\right )}{5 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 79, normalized size = 1.05 \[ \frac {2}{5 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c\right )}} - \frac {2 \, x}{5 \, \sqrt {-a^{2} c x^{2} + c} c^{2}} - \frac {x}{5 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 56, normalized size = 0.75 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3+4\,a^2\,x^2+a\,x-2\right )}{5\,a\,c^3\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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