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])]/(Sq
rt[a^2 - 4*b*c]*d^2*Log[f]^2)

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Rubi [A]  time = 0.63392, 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 \[ \frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}-\frac{\text{PolyLog}\left (2,-\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}\right )}{d^2 \log ^2(f) \sqrt{a^2-4 b c}}+\frac{x \log \left (\frac{2 c f^{c+d x}}{a-\sqrt{a^2-4 b c}}+1\right )}{d \log (f) \sqrt{a^2-4 b c}}-\frac{x \log \left (\frac{2 c f^{c+d x}}{\sqrt{a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt{a^2-4 b c}} \]

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])]/(Sq
rt[a^2 - 4*b*c]*d^2*Log[f]^2)

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Rubi in Sympy [A]  time = 71.1219, size = 185, normalized size = 0.91 \[ \frac{x \log{\left (\frac{2 c f^{c + d x}}{a - \sqrt{a^{2} - 4 b c}} + 1 \right )}}{d \sqrt{a^{2} - 4 b c} \log{\left (f \right )}} - \frac{x \log{\left (\frac{2 c f^{c + d x}}{a + \sqrt{a^{2} - 4 b c}} + 1 \right )}}{d \sqrt{a^{2} - 4 b c} \log{\left (f \right )}} + \frac{\operatorname{Li}_{2}\left (- \frac{2 c f^{c + d x}}{a - \sqrt{a^{2} - 4 b c}}\right )}{d^{2} \sqrt{a^{2} - 4 b c} \log{\left (f \right )}^{2}} - \frac{\operatorname{Li}_{2}\left (- \frac{2 c f^{c + d x}}{a + \sqrt{a^{2} - 4 b c}}\right )}{d^{2} \sqrt{a^{2} - 4 b c} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*f**(-d*x-c)+c*f**(d*x+c)),x)

[Out]

x*log(2*c*f**(c + d*x)/(a - sqrt(a**2 - 4*b*c)) + 1)/(d*sqrt(a**2 - 4*b*c)*log(f
)) - x*log(2*c*f**(c + d*x)/(a + sqrt(a**2 - 4*b*c)) + 1)/(d*sqrt(a**2 - 4*b*c)*
log(f)) + polylog(2, -2*c*f**(c + d*x)/(a - sqrt(a**2 - 4*b*c)))/(d**2*sqrt(a**2
- 4*b*c)*log(f)**2) - polylog(2, -2*c*f**(c + d*x)/(a + sqrt(a**2 - 4*b*c)))/(d
**2*sqrt(a**2 - 4*b*c)*log(f)**2)

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Mathematica [A]  time = 10.3683, 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 [C]  time = 0.048, size = 426, normalized size = 2.1 \[ -{\frac{x}{d\ln \left ( f \right ) }\ln \left ({1 \left ( -2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}+{\frac{x}{d\ln \left ( f \right ) }\ln \left ({1 \left ( 2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}-{\frac{c}{\ln \left ( f \right ){d}^{2}}\ln \left ({1 \left ( -2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}+{\frac{c}{\ln \left ( f \right ){d}^{2}}\ln \left ({1 \left ( 2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}+{\frac{1}{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it dilog} \left ({1 \left ( -2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}-a \right ) \left ( -a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}-{\frac{1}{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it dilog} \left ({1 \left ( 2\,b{f}^{-dx-c}+\sqrt{{a}^{2}-4\,cb}+a \right ) \left ( a+\sqrt{{a}^{2}-4\,cb} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}+2\,{\frac{c}{\ln \left ( f \right ){d}^{2}\sqrt{-{a}^{2}+4\,cb}}\arctan \left ({\frac{2\,b{f}^{-dx-c}+a}{\sqrt{-{a}^{2}+4\,cb}}} \right ) } \]

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-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-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-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-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-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-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-c)+a)/(-
a^2+4*b*c)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305017, size = 558, normalized size = 2.75 \[ \frac{b c \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) - b c \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) -{\left (b d x + b c\right )} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (\frac{2 \, c f^{d x + c} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}\right ) +{\left (b d x + b c\right )} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (-\frac{2 \, c f^{d x + c} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a}\right ) - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (-\frac{2 \, c f^{d x + c} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a} + 1\right ) + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (\frac{2 \, c f^{d x + c} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a} + 1\right )}{{\left (a^{2} - 4 \, b c\right )} d^{2} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*f^(d*x + c) + b*f^(-d*x - c) + a),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)*log(f)*log((2*c*f^(d*
x + c) + b*sqrt((a^2 - 4*b*c)/b^2) + a)/(b*sqrt((a^2 - 4*b*c)/b^2) + a)) + (b*d*
x + b*c)*sqrt((a^2 - 4*b*c)/b^2)*log(f)*log(-(2*c*f^(d*x + c) - b*sqrt((a^2 - 4*
b*c)/b^2) + a)/(b*sqrt((a^2 - 4*b*c)/b^2) - a)) - b*sqrt((a^2 - 4*b*c)/b^2)*dilo
g(-(2*c*f^(d*x + c) + b*sqrt((a^2 - 4*b*c)/b^2) + a)/(b*sqrt((a^2 - 4*b*c)/b^2)
+ a) + 1) + b*sqrt((a^2 - 4*b*c)/b^2)*dilog((2*c*f^(d*x + c) - b*sqrt((a^2 - 4*b
*c)/b^2) + a)/(b*sqrt((a^2 - 4*b*c)/b^2) - a) + 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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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