Optimal. Leaf size=46 \[ -\frac {\sqrt {1-a^2 x^2}}{2 a c (1+a x)^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.07, 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 = {6278, 6275, 32}
\begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{2 a c (a x+1)^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1+a x)^3} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 a c (1+a x)^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 1.15 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{2 a c^2 (-1+a x) (1+a x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 53, normalized size = 1.15
method | result | size |
gosper | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 \left (a x +1\right )^{2} a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(38\) |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}}{2 \left (a^{2} x^{2}-1\right ) c^{2} a \left (a x +1\right )^{2}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.63 \begin {gather*} -\frac {1}{2 \, {\left (a^{3} c^{\frac {3}{2}} x^{2} + 2 \, a^{2} c^{\frac {3}{2}} x + a c^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 72, normalized size = 1.57 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x^{2} + 2 \, x\right )}}{2 \, {\left (a^{4} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{3} - 2 \, a c^{2} x - c^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 58, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^2\,\left (\frac {2\,x}{a^3}+\frac {1}{a^4}-x^4-\frac {2\,x^3}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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