Integrand size = 20, antiderivative size = 38 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {4}{3 a (c-a c x)^{3/2}}+\frac {2}{a c \sqrt {c-a c x}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6265, 21, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {2}{a c \sqrt {c-a c x}}-\frac {4}{3 a (c-a c x)^{3/2}} \]
[In]
[Out]
Rule 21
Rule 45
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx \\ & = -\int \frac {1+a x}{(1-a x) (c-a c x)^{3/2}} \, dx \\ & = -\left (c \int \frac {1+a x}{(c-a c x)^{5/2}} \, dx\right ) \\ & = -\left (c \int \left (\frac {2}{(c-a c x)^{5/2}}-\frac {1}{c (c-a c x)^{3/2}}\right ) \, dx\right ) \\ & = -\frac {4}{3 a (c-a c x)^{3/2}}+\frac {2}{a c \sqrt {c-a c x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {-2+6 a x}{3 a c (-1+a x) \sqrt {c-a c x}} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {2 \left (3 a x -1\right )}{3 a \left (-a c x +c \right )^{\frac {3}{2}}}\) | \(21\) |
default | \(\frac {\frac {2}{\sqrt {-a c x +c}}-\frac {4 c}{3 \left (-a c x +c \right )^{\frac {3}{2}}}}{a c}\) | \(31\) |
trager | \(-\frac {2 \left (3 a x -1\right ) \sqrt {-a c x +c}}{3 c^{2} \left (a x -1\right )^{2} a}\) | \(31\) |
pseudoelliptic | \(\frac {6 a x -2}{3 a c \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}}\) | \(32\) |
derivativedivides | \(-\frac {2 \left (-\frac {1}{\sqrt {-a c x +c}}+\frac {2 c}{3 \left (-a c x +c \right )^{\frac {3}{2}}}\right )}{c a}\) | \(33\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {2 \, \sqrt {-a c x + c} {\left (3 \, a x - 1\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
[In]
[Out]
Time = 1.75 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 c}{3 \left (- a c x + c\right )^{\frac {3}{2}}} - \frac {1}{\sqrt {- a c x + c}}\right )}{a c} & \text {for}\: a c \neq 0 \\\frac {\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x + 2 \log {\left (a x - 1 \right )} - 1}{a} & \text {otherwise} \end {cases}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a c x - c\right )}}{3 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a c} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {2 \, {\left (3 \, a c x - c\right )}}{3 \, {\left (a c x - c\right )} \sqrt {-a c x + c} a c} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {6\,a\,x-2}{3\,a\,{\left (c-a\,c\,x\right )}^{3/2}} \]
[In]
[Out]