\(\int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx\) [134]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2241} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \]

[In]

Int[f^(a + b/x^2)/x,x]

[Out]

-1/2*(f^a*ExpIntegralEi[(b*Log[f])/x^2])

Rule 2241

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*} \text {integral}& = -\frac {1}{2} f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \]

[In]

Integrate[f^(a + b/x^2)/x,x]

[Out]

-1/2*(f^a*ExpIntegralEi[(b*Log[f])/x^2])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {f^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}\)	\(16\)
meijerg	\(-\frac {f^{a} \left (-2 \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )-\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )-\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )\right )}{2}\)	\(41\)

[In]

int(f^(a+b/x^2)/x,x,method=_RETURNVERBOSE)

[Out]

1/2*f^a*Ei(1,-b*ln(f)/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {1}{2} \, f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \]

[In]

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

[Out]

-1/2*f^a*Ei(b*log(f)/x^2)

Sympy [F]

\[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=\int \frac {f^{a + \frac {b}{x^{2}}}}{x}\, dx \]

[In]

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

[Out]

Integral(f**(a + b/x**2)/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {1}{2} \, f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \]

[In]

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

[Out]

-1/2*f^a*Ei(b*log(f)/x^2)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x^2)/x, x)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx=-\frac {f^a\,\mathrm {ei}\left (\frac {b\,\ln \left (f\right )}{x^2}\right )}{2} \]

[In]

int(f^(a + b/x^2)/x,x)

[Out]

-(f^a*ei((b*log(f))/x^2))/2