Integrand size = 16, antiderivative size = 65 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {3 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6310, 6313, 1821, 821, 272, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {3 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a^2 x^2}}+2 c x \sqrt {1-\frac {1}{a^2 x^2}} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left ((a c) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right ) \\ & = (a c) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{2} (a c) \text {Subst}\left (\int \frac {\frac {4}{a}-\frac {3 x}{a^2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = 2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(3 c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = 2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {(3 c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a} \\ & = 2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{2} (3 a c) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = 2 c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {3 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (-4+a x)+3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{2 a} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {\left (a x -4\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{2 a}-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(99\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -4 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+4 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\) | \(153\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} - 3 \, a c x - 4 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a} \]
[In]
[Out]
\[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=- c \left (\int a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (55) = 110\).
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.08 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {1}{2} \, a {\left (\frac {2 \, {\left (5 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {3 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} - \frac {1}{2} \, \sqrt {a^{2} x^{2} - 1} {\left (c x \mathrm {sgn}\left (a x + 1\right ) - \frac {4 \, c \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.48 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {3\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}-5\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {3\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
[In]
[Out]