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

Optimal. Leaf size=15 \[ -\frac{1}{3} f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right ) \]

[Out]

-(f^a*ExpIntegralEi[(b*Log[f])/x^3])/3

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Rubi [A]  time = 0.0353709, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{3} f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^3)/x,x]

[Out]

-(f^a*ExpIntegralEi[(b*Log[f])/x^3])/3

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Rubi in Sympy [A]  time = 3.06504, size = 15, normalized size = 1. \[ - \frac{f^{a} \operatorname{Ei}{\left (\frac{b \log{\left (f \right )}}{x^{3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**a*Ei(b*log(f)/x**3)/3

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Mathematica [A]  time = 0.00514245, size = 15, normalized size = 1. \[ -\frac{1}{3} f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^3)/x,x]

[Out]

-(f^a*ExpIntegralEi[(b*Log[f])/x^3])/3

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Maple [B]  time = 0.023, size = 41, normalized size = 2.7 \[ -{\frac{{f}^{a}}{3} \left ( -3\,\ln \left ( x \right ) +\ln \left ( -b \right ) +\ln \left ( \ln \left ( f \right ) \right ) -\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) -{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^3)/x,x)

[Out]

-1/3*f^a*(-3*ln(x)+ln(-b)+ln(ln(f))-ln(-b*ln(f)/x^3)-Ei(1,-b*ln(f)/x^3))

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Maxima [A]  time = 0.802509, size = 18, normalized size = 1.2 \[ -\frac{1}{3} \, f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*f^a*Ei(b*log(f)/x^3)

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Fricas [A]  time = 0.270911, size = 18, normalized size = 1.2 \[ -\frac{1}{3} \, f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*f^a*Ei(b*log(f)/x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**3)/x,x)

[Out]

Integral(f**(a + b/x**3)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(a + b/x^3)/x, x)