Integrand size = 13, antiderivative size = 69 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {1}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x}{a^3}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2320, 46} \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac {x}{a^3}+\frac {1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {1}{2 a d \left (a+b e^{c+d x}\right )^2} \]
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Rule 46
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {1}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x}{a^3}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {\frac {a \left (3 a+2 b e^{c+d x}\right )}{\left (a+b e^{c+d x}\right )^2}+2 \log \left (e^{c+d x}\right )-2 \log \left (a+b e^{c+d x}\right )}{2 a^3 d} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a^{3}}}{d}\) | \(66\) |
default | \(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a^{3}}}{d}\) | \(66\) |
risch | \(\frac {x}{a^{3}}+\frac {c}{a^{3} d}+\frac {2 b \,{\mathrm e}^{d x +c}+3 a}{2 a^{2} d \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a}{b}\right )}{a^{3} d}\) | \(69\) |
norman | \(\frac {\frac {x}{a}+\frac {b^{2} x \,{\mathrm e}^{2 d x +2 c}}{a^{3}}+\frac {2 b x \,{\mathrm e}^{d x +c}}{a^{2}}-\frac {2 b \,{\mathrm e}^{d x +c}}{a^{2} d}-\frac {3 b^{2} {\mathrm e}^{2 d x +2 c}}{2 a^{3} d}}{\left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3} d}\) | \(103\) |
parallelrisch | \(-\frac {-2 \,{\mathrm e}^{2 d x +2 c} b^{2} d x +2 \ln \left (a +b \,{\mathrm e}^{d x +c}\right ) {\mathrm e}^{2 d x +2 c} b^{2}-4 \,{\mathrm e}^{d x +c} a b d x +4 \ln \left (a +b \,{\mathrm e}^{d x +c}\right ) {\mathrm e}^{d x +c} a b -2 a^{2} d x +3 b^{2} {\mathrm e}^{2 d x +2 c}+2 \ln \left (a +b \,{\mathrm e}^{d x +c}\right ) a^{2}+4 \,{\mathrm e}^{d x +c} a b}{2 a^{3} d \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}\) | \(140\) |
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Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {2 \, b^{2} d x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d x + 3 \, a^{2} + 2 \, {\left (2 \, a b d x + a b\right )} e^{\left (d x + c\right )} - 2 \, {\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \, {\left (a^{3} b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d e^{\left (d x + c\right )} + a^{5} d\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {3 a + 2 b e^{c + d x}}{2 a^{4} d + 4 a^{3} b d e^{c + d x} + 2 a^{2} b^{2} d e^{2 c + 2 d x}} + \frac {x}{a^{3}} - \frac {\log {\left (\frac {a}{b} + e^{c + d x} \right )}}{a^{3} d} \]
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Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {2 \, b e^{\left (d x + c\right )} + 3 \, a}{2 \, {\left (a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b e^{\left (d x + c\right )} + a^{4}\right )} d} + \frac {d x + c}{a^{3} d} - \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d} \]
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Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {\frac {2 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, \log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a^{3}} + \frac {2 \, a b e^{\left (d x + c\right )} + 3 \, a^{2}}{{\left (b e^{\left (d x + c\right )} + a\right )}^{2} a^{3}}}{2 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {\frac {x}{a}+\frac {b^2\,x\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^3}+\frac {2\,b\,x\,{\mathrm {e}}^{c+d\,x}}{a^2}-\frac {3\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{2\,a^3\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2\,d}}{a^2+2\,{\mathrm {e}}^{c+d\,x}\,a\,b+{\mathrm {e}}^{2\,c+2\,d\,x}\,b^2}-\frac {\ln \left (a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d} \]
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