Optimal. Leaf size=59 \[ \frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {2 (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6167, 6141, 653, 217, 203} \[ \frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {2 (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 653
Rule 6141
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\left (c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\right )\\ &=-\frac {2 (1+a x)}{a \sqrt {c-a^2 c x^2}}+\int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {2 (1+a x)}{a \sqrt {c-a^2 c x^2}}+\operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=-\frac {2 (1+a x)}{a \sqrt {c-a^2 c x^2}}+\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 1.39 \[ -\frac {2 \sqrt {1-a^2 x^2} \left (\sqrt {a x+1}+\sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-a x} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 153, normalized size = 2.59 \[ \left [-\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x - a c\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x - a c}\right ] \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 1.34 \[ \frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{2} c \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 40, normalized size = 0.68 \[ \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x - a c} + \frac {\arcsin \left (a x\right )}{a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a\,x+1}{\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )} \,d x \]
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 \[ \int \frac {a x + 1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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