Optimal. Leaf size=110 \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {3 x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {3 x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(1-a x)^2}{1+a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (-3+a x+\frac {4}{1+a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.48 \[ \frac {\sqrt {c-a^2 c x^2} \left (\frac {a x^2}{2}+\frac {4 \log (a x+1)}{a}-3 x\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 347, normalized size = 3.15 \[ \left [\frac {4 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 6 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a^{3} x^{2} - a\right )}}, \frac {8 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 6 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a^{3} x^{2} - a\right )}}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 0.57 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-6 a x +8 \ln \left (a x +1\right )\right )}{2 \left (a^{2} x^{2}-1\right ) a} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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