Optimal. Leaf size=62 \[ -\frac {\log \left (a e^{2 p x}+b\right )}{4 a b p^2}+\frac {x}{2 a b p}-\frac {x}{2 a p \left (a e^{2 p x}+b\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2283, 2191, 2282, 36, 29, 31} \[ -\frac {\log \left (a e^{2 p x}+b\right )}{4 a b p^2}+\frac {x}{2 a b p}-\frac {x}{2 a p \left (a e^{2 p x}+b\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2191
Rule 2282
Rule 2283
Rubi steps
\begin {align*} \int \frac {x}{\left (b e^{-p x}+a e^{p x}\right )^2} \, dx &=\int \frac {e^{2 p x} x}{\left (b+a e^{2 p x}\right )^2} \, dx\\ &=-\frac {x}{2 a \left (b+a e^{2 p x}\right ) p}+\frac {\int \frac {1}{b+a e^{2 p x}} \, dx}{2 a p}\\ &=-\frac {x}{2 a \left (b+a e^{2 p x}\right ) p}+\frac {\operatorname {Subst}\left (\int \frac {1}{x (b+a x)} \, dx,x,e^{2 p x}\right )}{4 a p^2}\\ &=-\frac {x}{2 a \left (b+a e^{2 p x}\right ) p}-\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x} \, dx,x,e^{2 p x}\right )}{4 b p^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 p x}\right )}{4 a b p^2}\\ &=\frac {x}{2 a b p}-\frac {x}{2 a \left (b+a e^{2 p x}\right ) p}-\frac {\log \left (b+a e^{2 p x}\right )}{4 a b p^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 49, normalized size = 0.79 \[ \frac {\frac {2 p x e^{2 p x}}{a e^{2 p x}+b}-\frac {\log \left (a e^{2 p x}+b\right )}{a}}{4 b p^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 58, normalized size = 0.94 \[ \frac {2 \, a p x e^{\left (2 \, p x\right )} - {\left (a e^{\left (2 \, p x\right )} + b\right )} \log \left (a e^{\left (2 \, p x\right )} + b\right )}{4 \, {\left (a^{2} b p^{2} e^{\left (2 \, p x\right )} + a b^{2} p^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.21, size = 74, normalized size = 1.19 \[ \frac {2 \, a p x e^{\left (2 \, p x\right )} - a e^{\left (2 \, p x\right )} \log \left (-a e^{\left (2 \, p x\right )} - b\right ) - b \log \left (-a e^{\left (2 \, p x\right )} - b\right )}{4 \, {\left (a^{2} b p^{2} e^{\left (2 \, p x\right )} + a b^{2} p^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 51, normalized size = 0.82 \[ \frac {x \,{\mathrm e}^{2 p x}}{2 \left (a \,{\mathrm e}^{2 p x}+b \right ) b p}-\frac {\ln \left (a \,{\mathrm e}^{2 p x}+b \right )}{4 a b \,p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 51, normalized size = 0.82 \[ \frac {x e^{\left (2 \, p x\right )}}{2 \, {\left (a b p e^{\left (2 \, p x\right )} + b^{2} p\right )}} - \frac {\log \left (\frac {a e^{\left (2 \, p x\right )} + b}{a}\right )}{4 \, a b p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 47, normalized size = 0.76 \[ \frac {x\,{\mathrm {e}}^{2\,p\,x}}{2\,b\,p\,\left (b+a\,{\mathrm {e}}^{2\,p\,x}\right )}-\frac {\ln \left (b+a\,{\mathrm {e}}^{2\,p\,x}\right )}{4\,a\,b\,p^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 51, normalized size = 0.82 \[ \frac {x}{2 a b p + 2 b^{2} p e^{- 2 p x}} - \frac {x}{2 a b p} - \frac {\log {\left (\frac {a}{b} + e^{- 2 p x} \right )}}{4 a b p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________