Optimal. Leaf size=61 \[ \frac {2 b \log \left (a+b e^{n x}\right )}{a^3 n}-\frac {2 b x}{a^3}-\frac {b}{a^2 n \left (a+b e^{n x}\right )}-\frac {e^{-n x}}{a^2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2248, 44} \[ -\frac {b}{a^2 n \left (a+b e^{n x}\right )}+\frac {2 b \log \left (a+b e^{n x}\right )}{a^3 n}-\frac {2 b x}{a^3}-\frac {e^{-n x}}{a^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{-n x}}{\left (a+b e^{n x}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac {e^{-n x}}{a^2 n}-\frac {b}{a^2 \left (a+b e^{n x}\right ) n}-\frac {2 b x}{a^3}+\frac {2 b \log \left (a+b e^{n x}\right )}{a^3 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 49, normalized size = 0.80 \[ -\frac {a \left (\frac {b}{a+b e^{n x}}+e^{-n x}\right )-2 b \log \left (a+b e^{n x}\right )+2 b n x}{a^3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 84, normalized size = 1.38 \[ -\frac {2 \, b^{2} n x e^{\left (2 \, n x\right )} + a^{2} + 2 \, {\left (a b n x + a b\right )} e^{\left (n x\right )} - 2 \, {\left (b^{2} e^{\left (2 \, n x\right )} + a b e^{\left (n x\right )}\right )} \log \left (b e^{\left (n x\right )} + a\right )}{a^{3} b n e^{\left (2 \, n x\right )} + a^{4} n e^{\left (n x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 59, normalized size = 0.97 \[ -\frac {\frac {2 \, b n x}{a^{3}} - \frac {2 \, b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{3}} + \frac {2 \, b e^{\left (n x\right )} + a}{{\left (b e^{\left (2 \, n x\right )} + a e^{\left (n x\right )}\right )} a^{2}}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 67, normalized size = 1.10 \[ -\frac {b}{\left (b \,{\mathrm e}^{n x}+a \right ) a^{2} n}-\frac {{\mathrm e}^{-n x}}{a^{2} n}+\frac {2 b \ln \left (b \,{\mathrm e}^{n x}+a \right )}{a^{3} n}-\frac {2 b \ln \left ({\mathrm e}^{n x}\right )}{a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 57, normalized size = 0.93 \[ \frac {b^{2}}{{\left (a^{4} e^{\left (-n x\right )} + a^{3} b\right )} n} - \frac {e^{\left (-n x\right )}}{a^{2} n} + \frac {2 \, b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.67, size = 86, normalized size = 1.41 \[ \frac {2\,b\,\ln \left (a+b\,{\mathrm {e}}^{n\,x}\right )}{a^3\,n}-\frac {\frac {1}{a\,n}+\frac {2\,b^2\,x\,{\mathrm {e}}^{2\,n\,x}}{a^3}-\frac {2\,b^2\,{\mathrm {e}}^{2\,n\,x}}{a^3\,n}+\frac {2\,b\,x\,{\mathrm {e}}^{n\,x}}{a^2}}{a\,{\mathrm {e}}^{n\,x}+b\,{\mathrm {e}}^{2\,n\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.19, size = 61, normalized size = 1.00 \[ \frac {b^{2}}{a^{4} n e^{- n x} + a^{3} b n} + \begin {cases} - \frac {e^{- n x}}{a^{2} n} & \text {for}\: a^{2} n \neq 0 \\\frac {x}{a^{2}} & \text {otherwise} \end {cases} + \frac {2 b \log {\left (e^{- n x} + \frac {b}{a} \right )}}{a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________