Optimal. Leaf size=69 \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6143, 6140} \[ \frac {a x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {x \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{\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 (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {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}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.58 \[ \frac {\left (\frac {a x^2}{2}+x\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 47, normalized size = 0.68 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x^{2} + 2 \, x\right )}}{2 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 0.49 \[ \frac {x \left (a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{2 \sqrt {-a^{2} x^{2}+1}} \]
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 = 0.95, size = 34, normalized size = 0.49 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a\,x^2}{2}+x\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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