Integrand size = 22, antiderivative size = 68 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6327, 6328} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.60 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {(2+a x) \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(\frac {x \left (a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(44\) |
default | \(\frac {\left (a x +2\right ) x \sqrt {-c \left (a^{2} x^{2}-1\right )}}{2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(45\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.32 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {-a^{2} c} {\left (a x^{2} + 2 \, x\right )}}{2 \, a} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
[In]
[Out]