3.541 \(\int \frac{x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx\)

Optimal. Leaf size=203 \[ \frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}-\frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}+\frac{x \log \left (\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}+1\right )}{d \log (f) \sqrt{a^2-4 b c}}-\frac{x \log \left (\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt{a^2-4 b c}} \]

[Out]

(x*Log[1 + (2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) - (x*Log[1 + (2*c*f^(c + d
*x))/(a + Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) + PolyLog[2, (-2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*
b*c])]/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2) - PolyLog[2, (-2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c])]/(Sqrt[a^2 - 4
*b*c]*d^2*Log[f]^2)

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Rubi [A]  time = 0.406733, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2267, 2264, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}-\frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}+\frac{x \log \left (\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}+1\right )}{d \log (f) \sqrt{a^2-4 b c}}-\frac{x \log \left (\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt{a^2-4 b c}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*f^(-c - d*x) + c*f^(c + d*x)),x]

[Out]

(x*Log[1 + (2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) - (x*Log[1 + (2*c*f^(c + d
*x))/(a + Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) + PolyLog[2, (-2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*
b*c])]/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2) - PolyLog[2, (-2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c])]/(Sqrt[a^2 - 4
*b*c]*d^2*Log[f]^2)

Rule 2267

Int[(u_)/((a_) + (b_.)*(F_)^(v_) + (c_.)*(F_)^(w_)), x_Symbol] :> Int[(u*F^v)/(c + a*F^v + b*F^(2*v)), x] /; F
reeQ[{F, a, b, c}, x] && EqQ[w, -v] && LinearQ[v, x] && If[RationalQ[Coefficient[v, x, 1]], GtQ[Coefficient[v,
x, 1], 0], LtQ[LeafCount[v], LeafCount[w]]]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx &=\int \frac{f^{c+d x} x}{b+a f^{c+d x}+c f^{2 (c+d x)}} \, dx\\ &=\frac{(2 c) \int \frac{f^{c+d x} x}{a-\sqrt{a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt{a^2-4 b c}}-\frac{(2 c) \int \frac{f^{c+d x} x}{a+\sqrt{a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt{a^2-4 b c}}\\ &=\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}-\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a+\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}-\frac{\int \log \left (1+\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right ) \, dx}{\sqrt{a^2-4 b c} d \log (f)}+\frac{\int \log \left (1+\frac{2 c f^{c+d x}}{a+\sqrt{a^2-4 b c}}\right ) \, dx}{\sqrt{a^2-4 b c} d \log (f)}\\ &=\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}-\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a+\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{a-\sqrt{a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt{a^2-4 b c} d^2 \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{a+\sqrt{a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt{a^2-4 b c} d^2 \log ^2(f)}\\ &=\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}-\frac{x \log \left (1+\frac{2 c f^{c+d x}}{a+\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d \log (f)}+\frac{\text{Li}_2\left (-\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d^2 \log ^2(f)}-\frac{\text{Li}_2\left (-\frac{2 c f^{c+d x}}{a+\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c} d^2 \log ^2(f)}\\ \end{align*}

Mathematica [F]  time = 0.450121, size = 0, normalized size = 0. \[ \int \frac{x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x/(a + b*f^(-c - d*x) + c*f^(c + d*x)),x]

[Out]

Integrate[x/(a + b*f^(-c - d*x) + c*f^(c + d*x)), x]

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Maple [B]  time = 0.067, size = 433, normalized size = 2.1 \begin{align*}{\frac{x}{d\ln \left ( f \right ) }\ln \left ({ \left ( 2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}-{\frac{x}{d\ln \left ( f \right ) }\ln \left ({ \left ( -2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}+{\frac{c}{\ln \left ( f \right ){d}^{2}}\ln \left ({ \left ( 2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}-{\frac{c}{\ln \left ( f \right ){d}^{2}}\ln \left ({ \left ( -2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}+{\frac{1}{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it dilog} \left ({ \left ( -2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}-{\frac{1}{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it dilog} \left ({ \left ( 2\,b{f}^{-dx}{f}^{-c}+\sqrt{{a}^{2}-4\,bc}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,bc} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,bc}}}}+2\,{\frac{c}{\ln \left ( f \right ){d}^{2}\sqrt{-{a}^{2}+4\,bc}}\arctan \left ({\frac{2\,b{f}^{-dx}{f}^{-c}+a}{\sqrt{-{a}^{2}+4\,bc}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*f^(-d*x-c)+c*f^(d*x+c)),x)

[Out]

1/ln(f)/d/(a^2-4*b*c)^(1/2)*ln((2*b*f^(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)+a)/(a+(a^2-4*b*c)^(1/2)))*x-1/ln(f)/d/(a
^2-4*b*c)^(1/2)*ln((-2*b*f^(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)-a)/(-a+(a^2-4*b*c)^(1/2)))*x+1/ln(f)/d^2/(a^2-4*b*c
)^(1/2)*ln((2*b*f^(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)+a)/(a+(a^2-4*b*c)^(1/2)))*c-1/ln(f)/d^2/(a^2-4*b*c)^(1/2)*ln
((-2*b*f^(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)-a)/(-a+(a^2-4*b*c)^(1/2)))*c+1/ln(f)^2/d^2/(a^2-4*b*c)^(1/2)*dilog((-
2*b*f^(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)-a)/(-a+(a^2-4*b*c)^(1/2)))-1/ln(f)^2/d^2/(a^2-4*b*c)^(1/2)*dilog((2*b*f^
(-d*x)*f^(-c)+(a^2-4*b*c)^(1/2)+a)/(a+(a^2-4*b*c)^(1/2)))+2/ln(f)/d^2*c/(-a^2+4*b*c)^(1/2)*arctan((2*b*f^(-d*x
)*f^(-c)+a)/(-a^2+4*b*c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*f^(-d*x-c)+c*f^(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3956, size = 845, normalized size = 4.16 \begin{align*} \frac{b c \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) - b c \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) +{\left (b d x + b c\right )} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (\frac{{\left (b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b}\right ) -{\left (b d x + b c\right )} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (-\frac{{\left (b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b}\right ) + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (-\frac{{\left (b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b} + 1\right ) - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (\frac{{\left (b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b} + 1\right )}{{\left (a^{2} - 4 \, b c\right )} d^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*f^(-d*x-c)+c*f^(d*x+c)),x, algorithm="fricas")

[Out]

(b*c*sqrt((a^2 - 4*b*c)/b^2)*log(2*c*f^(d*x + c) + b*sqrt((a^2 - 4*b*c)/b^2) + a)*log(f) - b*c*sqrt((a^2 - 4*b
*c)/b^2)*log(2*c*f^(d*x + c) - b*sqrt((a^2 - 4*b*c)/b^2) + a)*log(f) + (b*d*x + b*c)*sqrt((a^2 - 4*b*c)/b^2)*l
og(f)*log(1/2*((b*sqrt((a^2 - 4*b*c)/b^2) + a)*f^(d*x + c) + 2*b)/b) - (b*d*x + b*c)*sqrt((a^2 - 4*b*c)/b^2)*l
og(f)*log(-1/2*((b*sqrt((a^2 - 4*b*c)/b^2) - a)*f^(d*x + c) - 2*b)/b) + b*sqrt((a^2 - 4*b*c)/b^2)*dilog(-1/2*(
(b*sqrt((a^2 - 4*b*c)/b^2) + a)*f^(d*x + c) + 2*b)/b + 1) - b*sqrt((a^2 - 4*b*c)/b^2)*dilog(1/2*((b*sqrt((a^2
- 4*b*c)/b^2) - a)*f^(d*x + c) - 2*b)/b + 1))/((a^2 - 4*b*c)*d^2*log(f)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*f**(-d*x-c)+c*f**(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{c f^{d x + c} + b f^{-d x - c} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*f^(-d*x-c)+c*f^(d*x+c)),x, algorithm="giac")

[Out]

integrate(x/(c*f^(d*x + c) + b*f^(-d*x - c) + a), x)