Integrand size = 18, antiderivative size = 27 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \csc ^{-1}(a x)}{a} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6312, 283, 222} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=c x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {c \csc ^{-1}(a x)}{a} \]
[In]
[Out]
Rule 222
Rule 283
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2} \\ & = c \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {c \csc ^{-1}(a x)}{a} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x+\arcsin \left (\frac {1}{a x}\right )\right )}{a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(25)=50\).
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33
method | result | size |
default | \(\frac {\left (a x -1\right ) c \left (\sqrt {a^{2} x^{2}-1}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a}\) | \(63\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int \frac {a}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-2 \, a {\left (\frac {c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {c \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) - \sqrt {a^{2} x^{2} - 1} c}{a \mathrm {sgn}\left (a x + 1\right )} \]
[In]
[Out]
Time = 3.87 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
[In]
[Out]