Integrand size = 24, antiderivative size = 59 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 (1+a x)}{a \sqrt {c-a^2 c x^2}}+\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \]
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Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6302, 6276, 667, 223, 209} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {2 (a x+1)}{a \sqrt {c-a^2 c x^2}} \]
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Rule 209
Rule 223
Rule 667
Rule 6276
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(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}}+\text {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 {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \left (\sqrt {1+a x}+\sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-a x} \sqrt {c-a^2 c x^2}} \]
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Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{a^{2} c \left (x -\frac {1}{a}\right )}\) | \(79\) |
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Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.59 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\left [-\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x - a c\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x - a c}\right ] \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {a x + 1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x - a c} + \frac {\arcsin \left (a x\right )}{a \sqrt {c}} \]
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Exception generated. \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {a\,x+1}{\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )} \,d x \]
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