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

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

[Out]

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

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Rubi [A]  time = 0.0223782, 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, Rules used = {2210} \[ -\frac{1}{3} f^a \text{Ei}\left (\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x,x]

[Out]

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

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{f^{a+\frac{b}{x^3}}}{x} \, dx &=-\frac{1}{3} f^a \text{Ei}\left (\frac{b \log (f)}{x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0020762, size = 15, normalized size = 1. \[ -\frac{1}{3} f^a \text{Ei}\left (\frac{b \log (f)}{x^3}\right ) \]

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 = 0.029, size = 41, normalized size = 2.7 \begin{align*} -{\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 ) } \end{align*}

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 = 1.27502, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{3} \, f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \end{align*}

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 = 1.70802, size = 36, normalized size = 2.4 \begin{align*} -\frac{1}{3} \, f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x^{3}}\right ) \end{align*}

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. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x}\,{d x} \end{align*}

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)