Optimal. Leaf size=46 \[ -\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (a x+1)^2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6192, 6193, 32} \[ -\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (a x+1)^2 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{(1+a x)^3} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 51, normalized size = 1.11 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a^2 c x^2}}{2 c^2 (a x-1) (a x+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 39, normalized size = 0.85 \[ -\frac {\sqrt {-a^{2} c}}{2 \, {\left (a^{4} c^{2} x^{2} + 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.85 \[ -\frac {\left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 58, normalized size = 1.26 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^2\,c\,\left (x\,\sqrt {c-a^2\,c\,x^2}+\frac {\sqrt {c-a^2\,c\,x^2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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