\(\int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx\) [147]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2242, 2235} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

[In]

Int[f^(a + b/x^2)/x^2,x]

[Out]

-1/2*(f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(Sqrt[b]*Sqrt[Log[f]])

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

[In]

Integrate[f^(a + b/x^2)/x^2,x]

[Out]

-1/2*(f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(Sqrt[b]*Sqrt[Log[f]])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72

method	result	size
meijerg	\(-\frac {f^{a} \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right ) \sqrt {\pi }}{2 \sqrt {b}\, \sqrt {\ln \left (f \right )}}\)	\(28\)
risch	\(-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{2 \sqrt {-b \ln \left (f \right )}}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right )}{2 \, b \log \left (f\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(f**(a + b/x**2)/x**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {\sqrt {\pi } f^{a} {\left (\operatorname {erf}\left (\sqrt {-\frac {b \log \left (f\right )}{x^{2}}}\right ) - 1\right )}}{2 \, x \sqrt {-\frac {b \log \left (f\right )}{x^{2}}}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(f^(a + b/x^2)/x^2, x)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{2\,\sqrt {b\,\ln \left (f\right )}} \]

[In]

int(f^(a + b/x^2)/x^2,x)

[Out]

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