Optimal. Leaf size=32 \[ -\frac {a^2 e^{-n x}}{n}+\frac {b^2 e^{n x}}{n}+2 a b x \]
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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 = {2280, 45}
\begin {gather*} -\frac {a^2 e^{-n x}}{n}+2 a b x+\frac {b^2 e^{n x}}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2280
Rubi steps
\begin {align*} \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 1.12 \begin {gather*} \frac {-a^2 e^{-n x}+b^2 e^{n x}-2 a b \log \left (e^{-n x}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 34, normalized size = 1.06
method | result | size |
risch | \(-\frac {a^{2} {\mathrm e}^{-n x}}{n}+\frac {b^{2} {\mathrm e}^{n x}}{n}+2 a b x\) | \(31\) |
derivativedivides | \(\frac {b^{2} {\mathrm e}^{n x}-a^{2} {\mathrm e}^{-n x}+2 b a \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(34\) |
default | \(\frac {b^{2} {\mathrm e}^{n x}-a^{2} {\mathrm e}^{-n x}+2 b a \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(34\) |
norman | \(\left (\frac {b^{2} {\mathrm e}^{2 n x}}{n}-\frac {a^{2}}{n}+2 a b x \,{\mathrm e}^{n x}\right ) {\mathrm e}^{-n x}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 30, normalized size = 0.94 \begin {gather*} 2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 34, normalized size = 1.06 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 39, normalized size = 1.22 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.67, size = 30, normalized size = 0.94 \begin {gather*} 2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 30, normalized size = 0.94 \begin {gather*} 2\,a\,b\,x-\frac {{\mathrm {e}}^{-n\,x}\,\left (a^2-b^2\,{\mathrm {e}}^{2\,n\,x}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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