3.48 \(\int \frac{f^x x}{\left (a+b f^{2 x}\right )^2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

[Out]

-ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (f^x*x)/(2*a*(a +
b*f^(2*x))*Log[f]) + (x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]
) - ((I/4)*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2) +
((I/4)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2)

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Rubi [A]  time = 0.271671, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[(f^x*x)/(a + b*f^(2*x))^2,x]

[Out]

-ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (f^x*x)/(2*a*(a +
b*f^(2*x))*Log[f]) + (x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]
) - ((I/4)*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2) +
((I/4)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2)

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Rubi in Sympy [A]  time = 47.1636, size = 156, normalized size = 0.91 \[ \frac{f^{x} x}{2 a \left (a + b f^{2 x}\right ) \log{\left (f \right )}} + \frac{x \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b} \log{\left (f \right )}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b} \log{\left (f \right )}^{2}} - \frac{i \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{4 a^{\frac{3}{2}} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{i \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{4 a^{\frac{3}{2}} \sqrt{b} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**x*x/(a+b*f**(2*x))**2,x)

[Out]

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

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Mathematica [A]  time = 0.15363, size = 271, normalized size = 1.58 \[ \frac{\frac{-\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \log ^2(f)}-\frac{i x \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \log (f)}+\frac{i x^2}{2 \sqrt{a}}}{2 \sqrt{b}}+\frac{\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \log ^2(f)}+\frac{i x \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \log (f)}-\frac{i x^2}{2 \sqrt{a}}}{2 \sqrt{b}}}{2 a}+\frac{x f^x}{2 a \log (f) \left (a+b f^{2 x}\right )}-\frac{\left (\frac{b f^{2 x}}{a}+1\right ) \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f) \left (a+b f^{2 x}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(f^x*x)/(a + b*f^(2*x))^2,x]

[Out]

-((1 + (b*f^(2*x))/a)*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*(a + b*f
^(2*x))*Log[f]^2) + (f^x*x)/(2*a*(a + b*f^(2*x))*Log[f]) + ((((I/2)*x^2)/Sqrt[a]
- (I*x*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Log[f]) - (I*PolyLog[2, ((-I)
*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Log[f]^2))/(2*Sqrt[b]) + (((-I/2)*x^2)/Sqrt[a]
+ (I*x*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Log[f]) + (I*PolyLog[2, (I*Sqr
t[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Log[f]^2))/(2*Sqrt[b]))/(2*a)

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Maple [C]  time = 0.052, size = 195, normalized size = 1.1 \[{\frac{{f}^{x}x}{2\,\ln \left ( f \right ) a \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) }}+{\frac{x}{4\,\ln \left ( f \right ) a}\ln \left ({1 \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{x}{4\,\ln \left ( f \right ) a}\ln \left ({1 \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}a}{\it dilog} \left ({1 \left ( -b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{1}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}a}{\it dilog} \left ({1 \left ( b{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}a}\arctan \left ({b{f}^{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x*x/(a+b*f^(2*x))^2,x)

[Out]

1/2/ln(f)/a*f^x*x/(a+b*(f^x)^2)+1/4/ln(f)/a*x/(-a*b)^(1/2)*ln((-b*f^x+(-a*b)^(1/
2))/(-a*b)^(1/2))-1/4/ln(f)/a*x/(-a*b)^(1/2)*ln((b*f^x+(-a*b)^(1/2))/(-a*b)^(1/2
))+1/4/ln(f)^2/a/(-a*b)^(1/2)*dilog((-b*f^x+(-a*b)^(1/2))/(-a*b)^(1/2))-1/4/ln(f
)^2/a/(-a*b)^(1/2)*dilog((b*f^x+(-a*b)^(1/2))/(-a*b)^(1/2))-1/2/ln(f)^2/a/(a*b)^
(1/2)*arctan(b*f^x/(a*b)^(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(f^x*x/(b*f^(2*x) + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2669, size = 509, normalized size = 2.96 \[ \frac{2 \, b f^{x} x \log \left (f\right ) +{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (-\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) -{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )}{\rm Li}_2\left (\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) -{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} + 2 \, a \sqrt{-\frac{b}{a}}\right ) +{\left (b f^{2 \, x} \sqrt{-\frac{b}{a}} + a \sqrt{-\frac{b}{a}}\right )} \log \left (2 \, b f^{x} - 2 \, a \sqrt{-\frac{b}{a}}\right ) +{\left (b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) -{\left (b f^{2 \, x} x \sqrt{-\frac{b}{a}} \log \left (f\right ) + a x \sqrt{-\frac{b}{a}} \log \left (f\right )\right )} \log \left (-\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right )}{4 \,{\left (a b^{2} f^{2 \, x} \log \left (f\right )^{2} + a^{2} b \log \left (f\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^x*x/(b*f^(2*x) + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{f^{x} x}{2 a^{2} \log{\left (f \right )} + 2 a b f^{2 x} \log{\left (f \right )}} + \frac{\int \left (- \frac{f^{x}}{a + b f^{2 x}}\right )\, dx + \int \frac{f^{x} x \log{\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**x*x/(a+b*f**(2*x))**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x}{{\left (b f^{2 \, x} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^x*x/(b*f^(2*x) + a)^2,x, algorithm="giac")

[Out]

integrate(f^x*x/(b*f^(2*x) + a)^2, x)