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

Optimal. Leaf size=49 \[ x f^{a+\frac{b}{x^2}}-\sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right ) \]

[Out]

f^(a + b/x^2)*x - Sqrt[b]*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Sqrt[Log[f
]]

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Rubi [A]  time = 0.0605577, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x f^{a+\frac{b}{x^2}}-\sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2),x]

[Out]

f^(a + b/x^2)*x - Sqrt[b]*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Sqrt[Log[f
]]

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Rubi in Sympy [A]  time = 5.37912, size = 44, normalized size = 0.9 \[ - \sqrt{\pi } \sqrt{b} f^{a} \sqrt{\log{\left (f \right )}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )} + f^{a + \frac{b}{x^{2}}} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2),x)

[Out]

-sqrt(pi)*sqrt(b)*f**a*sqrt(log(f))*erfi(sqrt(b)*sqrt(log(f))/x) + f**(a + b/x**
2)*x

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Mathematica [A]  time = 0.0131193, size = 49, normalized size = 1. \[ x f^{a+\frac{b}{x^2}}-\sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2),x]

[Out]

f^(a + b/x^2)*x - Sqrt[b]*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Sqrt[Log[f
]]

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Maple [A]  time = 0.021, size = 44, normalized size = 0.9 \[{f}^{a}{f}^{{\frac{b}{{x}^{2}}}}x-{{f}^{a}\ln \left ( f \right ) b\sqrt{\pi }{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2),x)

[Out]

f^a*f^(b/x^2)*x-f^a*ln(f)*b*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 0.912201, size = 68, normalized size = 1.39 \[ f^{a} f^{\frac{b}{x^{2}}} x - \frac{\sqrt{\pi } b f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )}{x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f^a*f^(b/x^2)*x - sqrt(pi)*b*f^a*(erf(sqrt(-b*log(f)/x^2)) - 1)*log(f)/(x*sqrt(-
b*log(f)/x^2))

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Fricas [A]  time = 0.269228, size = 74, normalized size = 1.51 \[ -\frac{\sqrt{\pi } b f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right ) - \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}} x}{\sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(pi)*b*f^a*erf(sqrt(-b*log(f))/x)*log(f) - sqrt(-b*log(f))*f^((a*x^2 + b)/
x^2)*x)/sqrt(-b*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**2),x)

[Out]

Integral(f**(a + b/x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(a + b/x^2), x)