Integrand size = 17, antiderivative size = 31 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=\frac {e^{-2 x}}{2}-\frac {3 e^{2 x}}{2}+\frac {e^{4 x}}{4}+3 x \]
[Out]
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2320, 272, 45} \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=3 x+\frac {e^{-2 x}}{2}-\frac {3 e^{2 x}}{2}+\frac {e^{4 x}}{4} \]
[In]
[Out]
Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{x^3} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^3}{x^2} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-3-\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{2 x}\right ) \\ & = \frac {e^{-2 x}}{2}-\frac {3 e^{2 x}}{2}+\frac {e^{4 x}}{4}+3 x \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=\frac {1}{4} e^{-2 x} \left (2-6 e^{4 x}+e^{6 x}\right )+3 \log \left (e^x\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74
method | result | size |
risch | \(3 x +\frac {{\mathrm e}^{4 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{-2 x}}{2}\) | \(23\) |
parts | \(3 x +\frac {{\mathrm e}^{4 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{-2 x}}{2}\) | \(23\) |
derivativedivides | \(\frac {{\mathrm e}^{4 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{2}+3 \ln \left ({\mathrm e}^{x}\right )+\frac {{\mathrm e}^{-2 x}}{2}\) | \(25\) |
default | \(\frac {{\mathrm e}^{4 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{2}+3 \ln \left ({\mathrm e}^{x}\right )+\frac {{\mathrm e}^{-2 x}}{2}\) | \(25\) |
norman | \(\left (-\frac {3 \,{\mathrm e}^{5 x}}{2}+\frac {{\mathrm e}^{7 x}}{4}+3 \,{\mathrm e}^{3 x} x +\frac {{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{-3 x}\) | \(30\) |
parallelrisch | \(-\frac {\left (-{\mathrm e}^{6 x}-2+6 \,{\mathrm e}^{4 x}-12 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{4}\) | \(30\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=\frac {1}{4} \, {\left (12 \, x e^{\left (2 \, x\right )} + e^{\left (6 \, x\right )} - 6 \, e^{\left (4 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=3 x + \frac {e^{4 x}}{4} - \frac {3 e^{2 x}}{2} + \frac {e^{- 2 x}}{2} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=-\frac {1}{4} \, {\left (6 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + 3 \, x + \frac {1}{2} \, e^{\left (-2 \, x\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=-\frac {1}{2} \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} + 3 \, x + \frac {1}{4} \, e^{\left (4 \, x\right )} - \frac {3}{2} \, e^{\left (2 \, x\right )} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int e^x \left (-e^{-x}+e^x\right )^3 \, dx=3\,x+\frac {{\mathrm {e}}^{-2\,x}}{2}-\frac {3\,{\mathrm {e}}^{2\,x}}{2}+\frac {{\mathrm {e}}^{4\,x}}{4} \]
[In]
[Out]