Optimal. Leaf size=41 \[ -\frac {2 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{3 a c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6135} \[ -\frac {2 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{3 a c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6135
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{3 a c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 1.56 \[ \frac {2 (2 a x-1) \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/4} \sqrt [4]{a x+1} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 55, normalized size = 1.34 \[ -\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2 a x -1\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{3 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 49, normalized size = 1.20 \[ \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {4\,x}{3\,c}-\frac {2}{3\,a\,c}\right )}{\sqrt {c-a^2\,c\,x^2}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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