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 = 0.0907172, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 2283
Rule 2191
Rule 2282
Rule 36
Rule 29
Rule 31
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 = 0.0648487, 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]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 51, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }{4\,{p}^{2}ba}}+{\frac{x \left ({{\rm e}^{px}} \right ) ^{2}}{2\,bp \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.955416, size = 69, normalized size = 1.11 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.1886, size = 135, normalized size = 2.18 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.158276, size = 51, normalized size = 0.82 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07893, size = 100, normalized size = 1.61 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]