Optimal. Leaf size=68 \[ \frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.11, antiderivative size = 68, 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 = {6192, 6193} \[ \frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int (1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.60 \[ \frac {(a x+2) \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 22, normalized size = 0.32 \[ \frac {\sqrt {-a^{2} c} {\left (a x^{2} + 2 \, x\right )}}{2 \, a} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 44, normalized size = 0.65 \[ \frac {x \left (a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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