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

Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

[Out]

f^x/(2*a*(a + b*f^(2*x))*Log[f]) + ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt
[b]*Log[f])

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Rubi [A]  time = 0.0740082, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}+\frac{f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[f^x/(a + b*f^(2*x))^2,x]

[Out]

f^x/(2*a*(a + b*f^(2*x))*Log[f]) + ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt
[b]*Log[f])

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Rubi in Sympy [A]  time = 10.5192, size = 48, normalized size = 0.81 \[ \frac{f^{x}}{2 a \left (a + b f^{2 x}\right ) \log{\left (f \right )}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b} \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**x/(2*a*(a + b*f**(2*x))*log(f)) + atan(sqrt(b)*f**x/sqrt(a))/(2*a**(3/2)*sqrt
(b)*log(f))

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Mathematica [A]  time = 0.0778586, size = 53, normalized size = 0.9 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{f^x}{a^2+a b f^{2 x}}}{2 \log (f)} \]

Antiderivative was successfully verified.

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

[Out]

(f^x/(a^2 + a*b*f^(2*x)) + ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(a^(3/2)*Sqrt[b]))/(2*L
og[f])

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Maple [A]  time = 0.046, size = 82, normalized size = 1.4 \[{\frac{{f}^{x}}{2\,\ln \left ( f \right ) a \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) }}-{\frac{1}{4\,\ln \left ( f \right ) a}\ln \left ({f}^{x}-{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{4\,\ln \left ( f \right ) a}\ln \left ({f}^{x}+{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x/(a+b*f^(2*x))^2,x)

[Out]

1/2/ln(f)/a*f^x/(a+b*(f^x)^2)-1/4/(-a*b)^(1/2)/a/ln(f)*ln(f^x-1/(-a*b)^(1/2)*a)+
1/4/(-a*b)^(1/2)/a/ln(f)*ln(f^x+1/(-a*b)^(1/2)*a)

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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/(b*f^(2*x) + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266697, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b f^{2 \, x} + a\right )} \log \left (\frac{2 \, a b f^{x} + \sqrt{-a b} b f^{2 \, x} - \sqrt{-a b} a}{b f^{2 \, x} + a}\right ) + 2 \, \sqrt{-a b} f^{x}}{4 \,{\left (\sqrt{-a b} a b f^{2 \, x} \log \left (f\right ) + \sqrt{-a b} a^{2} \log \left (f\right )\right )}}, -\frac{{\left (b f^{2 \, x} + a\right )} \arctan \left (\frac{a}{\sqrt{a b} f^{x}}\right ) - \sqrt{a b} f^{x}}{2 \,{\left (\sqrt{a b} a b f^{2 \, x} \log \left (f\right ) + \sqrt{a b} a^{2} \log \left (f\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((b*f^(2*x) + a)*log((2*a*b*f^x + sqrt(-a*b)*b*f^(2*x) - sqrt(-a*b)*a)/(b*f
^(2*x) + a)) + 2*sqrt(-a*b)*f^x)/(sqrt(-a*b)*a*b*f^(2*x)*log(f) + sqrt(-a*b)*a^2
*log(f)), -1/2*((b*f^(2*x) + a)*arctan(a/(sqrt(a*b)*f^x)) - sqrt(a*b)*f^x)/(sqrt
(a*b)*a*b*f^(2*x)*log(f) + sqrt(a*b)*a^2*log(f))]

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Sympy [A]  time = 0.536304, size = 53, normalized size = 0.9 \[ \frac{f^{x}}{2 a^{2} \log{\left (f \right )} + 2 a b f^{2 x} \log{\left (f \right )}} + \frac{\operatorname{RootSum}{\left (16 z^{2} a^{3} b + 1, \left ( i \mapsto i \log{\left (4 i a^{2} + f^{x} \right )} \right )\right )}}{\log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**x/(2*a**2*log(f) + 2*a*b*f**(2*x)*log(f)) + RootSum(16*_z**2*a**3*b + 1, Lamb
da(_i, _i*log(4*_i*a**2 + f**x)))/log(f)

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GIAC/XCAS [A]  time = 0.239208, size = 66, normalized size = 1.12 \[ \frac{\arctan \left (\frac{b f^{x}}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a{\rm ln}\left (f\right )} + \frac{f^{x}}{2 \,{\left (b f^{2 \, x} + a\right )} a{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(b*f^x/sqrt(a*b))/(sqrt(a*b)*a*ln(f)) + 1/2*f^x/((b*f^(2*x) + a)*a*ln(
f))