3.57 \(\int \frac{x^3}{b f^{-x}+a f^x} \, dx\)

Optimal. Leaf size=268 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]

[Out]

(x^3*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (((3*I)/2)*x^2*Po
lyLog[2, ((-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (((3*I)/2)*x^
2*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + ((3*I)*x*Pol
yLog[3, ((-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - ((3*I)*x*PolyL
og[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - ((3*I)*PolyLog[4, (
(-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^4) + ((3*I)*PolyLog[4, (I*Sq
rt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.3977, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b/f^x + a*f^x),x]

[Out]

(x^3*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (((3*I)/2)*x^2*Po
lyLog[2, ((-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (((3*I)/2)*x^
2*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + ((3*I)*x*Pol
yLog[3, ((-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - ((3*I)*x*PolyL
og[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - ((3*I)*PolyLog[4, (
(-I)*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^4) + ((3*I)*PolyLog[4, (I*Sq
rt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 102.562, size = 264, normalized size = 0.99 \[ - \frac{x^{3} \operatorname{atan}{\left (\frac{\sqrt{b} f^{- x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (f \right )}} - \frac{3 i x^{2} \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{3 i x^{2} \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} - \frac{3 i x \operatorname{Li}_{3}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} + \frac{3 i x \operatorname{Li}_{3}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} - \frac{3 i \operatorname{Li}_{4}\left (- \frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{4}} + \frac{3 i \operatorname{Li}_{4}\left (\frac{i \sqrt{b} f^{- x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b/(f**x)+a*f**x),x)

[Out]

-x**3*atan(sqrt(b)*f**(-x)/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)) - 3*I*x**2*polylog(
2, -I*sqrt(b)*f**(-x)/sqrt(a))/(2*sqrt(a)*sqrt(b)*log(f)**2) + 3*I*x**2*polylog(
2, I*sqrt(b)*f**(-x)/sqrt(a))/(2*sqrt(a)*sqrt(b)*log(f)**2) - 3*I*x*polylog(3, -
I*sqrt(b)*f**(-x)/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)**3) + 3*I*x*polylog(3, I*sqrt
(b)*f**(-x)/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)**3) - 3*I*polylog(4, -I*sqrt(b)*f**
(-x)/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)**4) + 3*I*polylog(4, I*sqrt(b)*f**(-x)/sqr
t(a))/(sqrt(a)*sqrt(b)*log(f)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0521668, size = 224, normalized size = 0.84 \[ \frac{i \left (-3 x^2 \log ^2(f) \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+3 x^2 \log ^2(f) \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-6 \text{PolyLog}\left (4,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+6 \text{PolyLog}\left (4,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+6 x \log (f) \text{PolyLog}\left (3,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-6 x \log (f) \text{PolyLog}\left (3,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+x^3 \log ^3(f) \log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-x^3 \log ^3(f) \log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )\right )}{2 \sqrt{a} \sqrt{b} \log ^4(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(b/f^x + a*f^x),x]

[Out]

((I/2)*(x^3*Log[f]^3*Log[1 - (I*Sqrt[a]*f^x)/Sqrt[b]] - x^3*Log[f]^3*Log[1 + (I*
Sqrt[a]*f^x)/Sqrt[b]] - 3*x^2*Log[f]^2*PolyLog[2, ((-I)*Sqrt[a]*f^x)/Sqrt[b]] +
3*x^2*Log[f]^2*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]] + 6*x*Log[f]*PolyLog[3, ((-I)
*Sqrt[a]*f^x)/Sqrt[b]] - 6*x*Log[f]*PolyLog[3, (I*Sqrt[a]*f^x)/Sqrt[b]] - 6*Poly
Log[4, ((-I)*Sqrt[a]*f^x)/Sqrt[b]] + 6*PolyLog[4, (I*Sqrt[a]*f^x)/Sqrt[b]]))/(Sq
rt[a]*Sqrt[b]*Log[f]^4)

_______________________________________________________________________________________

Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ({\frac{b}{{f}^{x}}}+a{f}^{x} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b/(f^x)+a*f^x),x)

[Out]

int(x^3/(b/(f^x)+a*f^x),x)

_______________________________________________________________________________________

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^3/(a*f^x + b/f^x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.255991, size = 433, normalized size = 1.62 \[ \frac{x^{3} \sqrt{-\frac{a}{b}} \log \left (f\right )^{3} \log \left (\frac{a f^{x} + b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}}\right ) - x^{3} \sqrt{-\frac{a}{b}} \log \left (f\right )^{3} \log \left (-\frac{a f^{x} - b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}}\right ) + 3 \, x^{2} \sqrt{-\frac{a}{b}}{\rm Li}_2\left (-\frac{a f^{x} + b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}} + 1\right ) \log \left (f\right )^{2} - 3 \, x^{2} \sqrt{-\frac{a}{b}}{\rm Li}_2\left (\frac{a f^{x} - b \sqrt{-\frac{a}{b}}}{b \sqrt{-\frac{a}{b}}} + 1\right ) \log \left (f\right )^{2} + 6 \, x \sqrt{-\frac{a}{b}} \log \left (f\right ){\rm Li}_{3}(\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) - 6 \, x \sqrt{-\frac{a}{b}} \log \left (f\right ){\rm Li}_{3}(-\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) - 6 \, \sqrt{-\frac{a}{b}}{\rm Li}_{4}(\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}}) + 6 \, \sqrt{-\frac{a}{b}}{\rm Li}_{4}(-\frac{a f^{x}}{b \sqrt{-\frac{a}{b}}})}{2 \, a \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a*f^x + b/f^x),x, algorithm="fricas")

[Out]

1/2*(x^3*sqrt(-a/b)*log(f)^3*log((a*f^x + b*sqrt(-a/b))/(b*sqrt(-a/b))) - x^3*sq
rt(-a/b)*log(f)^3*log(-(a*f^x - b*sqrt(-a/b))/(b*sqrt(-a/b))) + 3*x^2*sqrt(-a/b)
*dilog(-(a*f^x + b*sqrt(-a/b))/(b*sqrt(-a/b)) + 1)*log(f)^2 - 3*x^2*sqrt(-a/b)*d
ilog((a*f^x - b*sqrt(-a/b))/(b*sqrt(-a/b)) + 1)*log(f)^2 + 6*x*sqrt(-a/b)*log(f)
*polylog(3, a*f^x/(b*sqrt(-a/b))) - 6*x*sqrt(-a/b)*log(f)*polylog(3, -a*f^x/(b*s
qrt(-a/b))) - 6*sqrt(-a/b)*polylog(4, a*f^x/(b*sqrt(-a/b))) + 6*sqrt(-a/b)*polyl
og(4, -a*f^x/(b*sqrt(-a/b))))/(a*log(f)^4)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x^{3}}{a f^{2 x} + b}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b/(f**x)+a*f**x),x)

[Out]

Integral(f**x*x**3/(a*f**(2*x) + b), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{a f^{x} + \frac{b}{f^{x}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a*f^x + b/f^x),x, algorithm="giac")

[Out]

integrate(x^3/(a*f^x + b/f^x), x)