Optimal. Leaf size=48 \[ \frac {\log \left (a+b e^{c-d x}\right )}{a^2 d}+\frac {x}{a^2}-\frac {1}{a d \left (a+b e^{c-d x}\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2282, 44} \[ \frac {\log \left (a+b e^{c-d x}\right )}{a^2 d}+\frac {x}{a^2}-\frac {1}{a d \left (a+b e^{c-d x}\right )} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b e^{c-d x}\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,e^{c-d x}\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,e^{c-d x}\right )}{d}\\ &=-\frac {1}{a d \left (a+b e^{c-d x}\right )}+\frac {x}{a^2}+\frac {\log \left (a+b e^{c-d x}\right )}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 0.88 \[ \frac {\frac {b e^c}{a e^{d x}+b e^c}+\log \left (a e^{d x}+b e^c\right )}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 65, normalized size = 1.35 \[ \frac {b d x e^{\left (-d x + c\right )} + a d x + {\left (b e^{\left (-d x + c\right )} + a\right )} \log \left (b e^{\left (-d x + c\right )} + a\right ) - a}{a^{2} b d e^{\left (-d x + c\right )} + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 54, normalized size = 1.12 \[ -\frac {b {\left (\frac {\log \left ({\left | -\frac {a}{b e^{\left (-d x + c\right )} + a} + 1 \right |}\right )}{a^{2} b} + \frac {1}{{\left (b e^{\left (-d x + c\right )} + a\right )} a b}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 1.21 \[ -\frac {1}{\left (b \,{\mathrm e}^{-d x +c}+a \right ) a d}+\frac {\ln \left (b \,{\mathrm e}^{-d x +c}+a \right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{-d x +c}\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 55, normalized size = 1.15 \[ -\frac {1}{{\left (a b e^{\left (-d x + c\right )} + a^{2}\right )} d} + \frac {d x - c}{a^{2} d} + \frac {\log \left (b e^{\left (-d x + c\right )} + a\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 68, normalized size = 1.42 \[ \frac {\frac {x}{a}+\frac {b\,x\,{\mathrm {e}}^{c-d\,x}}{a^2}+\frac {b\,{\mathrm {e}}^{c-d\,x}}{a^2\,d}}{a+b\,{\mathrm {e}}^{c-d\,x}}+\frac {\ln \left (a+b\,{\mathrm {e}}^{-d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 39, normalized size = 0.81 \[ - \frac {1}{a^{2} d + a b d e^{c - d x}} + \frac {x}{a^{2}} + \frac {\log {\left (\frac {a}{b} + e^{c - d x} \right )}}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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