Optimal. Leaf size=19 \[ \frac {4}{\left (5-e^x\right )^2}+\frac {x}{-5+x} \]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 4, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6688, 6742, 2282, 44} \begin {gather*} \frac {4}{\left (5-e^x\right )^2}-\frac {5}{5-x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2282
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-625-75 e^{2 x}+5 e^{3 x}-e^x \left (-575+80 x-8 x^2\right )}{\left (5-e^x\right )^3 (5-x)^2} \, dx\\ &=\int \left (-\frac {40}{\left (-5+e^x\right )^3}-\frac {8}{\left (-5+e^x\right )^2}-\frac {5}{(-5+x)^2}\right ) \, dx\\ &=-\frac {5}{5-x}-8 \int \frac {1}{\left (-5+e^x\right )^2} \, dx-40 \int \frac {1}{\left (-5+e^x\right )^3} \, dx\\ &=-\frac {5}{5-x}-8 \operatorname {Subst}\left (\int \frac {1}{(-5+x)^2 x} \, dx,x,e^x\right )-40 \operatorname {Subst}\left (\int \frac {1}{(-5+x)^3 x} \, dx,x,e^x\right )\\ &=-\frac {5}{5-x}-8 \operatorname {Subst}\left (\int \left (\frac {1}{5 (-5+x)^2}-\frac {1}{25 (-5+x)}+\frac {1}{25 x}\right ) \, dx,x,e^x\right )-40 \operatorname {Subst}\left (\int \left (\frac {1}{5 (-5+x)^3}-\frac {1}{25 (-5+x)^2}+\frac {1}{125 (-5+x)}-\frac {1}{125 x}\right ) \, dx,x,e^x\right )\\ &=\frac {4}{\left (5-e^x\right )^2}-\frac {5}{5-x}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 17, normalized size = 0.89 \begin {gather*} \frac {4}{\left (-5+e^x\right )^2}+\frac {5}{-5+x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.85, size = 38, normalized size = 2.00 \begin {gather*} \frac {4 \, x + 5 \, e^{\left (2 \, x\right )} - 50 \, e^{x} + 105}{{\left (x - 5\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x - 5\right )} e^{x} + 25 \, x - 125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 44, normalized size = 2.32 \begin {gather*} \frac {4 \, x + 5 \, e^{\left (2 \, x\right )} - 50 \, e^{x} + 105}{x e^{\left (2 \, x\right )} - 10 \, x e^{x} + 25 \, x - 5 \, e^{\left (2 \, x\right )} + 50 \, e^{x} - 125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 17, normalized size = 0.89
method | result | size |
risch | \(\frac {5}{x -5}+\frac {4}{\left ({\mathrm e}^{x}-5\right )^{2}}\) | \(17\) |
norman | \(\frac {-50 \,{\mathrm e}^{x}+4 x +5 \,{\mathrm e}^{2 x}+105}{\left ({\mathrm e}^{x}-5\right )^{2} \left (x -5\right )}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.48, size = 38, normalized size = 2.00 \begin {gather*} \frac {4 \, x + 5 \, e^{\left (2 \, x\right )} - 50 \, e^{x} + 105}{{\left (x - 5\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x - 5\right )} e^{x} + 25 \, x - 125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.18, size = 27, normalized size = 1.42 \begin {gather*} \frac {4\,x+5\,{\mathrm {e}}^{2\,x}-50\,{\mathrm {e}}^x+105}{{\left ({\mathrm {e}}^x-5\right )}^2\,\left (x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 17, normalized size = 0.89 \begin {gather*} \frac {4}{e^{2 x} - 10 e^{x} + 25} + \frac {5}{x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________