Optimal. Leaf size=32 \[ -\frac {a^2 e^{-n x}}{n}+2 a b x+\frac {b^2 e^{n x}}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 32, 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, 43} \[ -\frac {a^2 e^{-n x}}{n}+2 a b x+\frac {b^2 e^{n x}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2248
Rubi steps
\begin {align*} \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac {a^2 e^{-n x}}{n}+\frac {b^2 e^{n x}}{n}+2 a b x\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 31, normalized size = 0.97 \[ \frac {a^2 \left (-e^{-n x}\right )+2 a b n x+b^2 e^{n x}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 34, normalized size = 1.06 \[ \frac {{\left (2 \, a b n x e^{\left (n x\right )} + b^{2} e^{\left (2 \, n x\right )} - a^{2}\right )} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 30, normalized size = 0.94 \[ 2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 39, normalized size = 1.22 \[ -\frac {a^{2} {\mathrm e}^{-n x}}{n}+\frac {2 a b \ln \left ({\mathrm e}^{n x}\right )}{n}+\frac {b^{2} {\mathrm e}^{n x}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.68, size = 30, normalized size = 0.94 \[ 2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 30, normalized size = 0.94 \[ 2\,a\,b\,x-\frac {{\mathrm {e}}^{-n\,x}\,\left (a^2-b^2\,{\mathrm {e}}^{2\,n\,x}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 39, normalized size = 1.22 \[ 2 a b x + \begin {cases} \frac {- a^{2} n e^{- n x} + b^{2} n e^{n x}}{n^{2}} & \text {for}\: n^{2} \neq 0 \\x \left (a^{2} + b^{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________