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 = 0.0342323, 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, Rules used = {2206, 2211, 2204} \[ x f^{a+\frac{b}{x^2}}-\sqrt{\pi } \sqrt{b} f^a \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right ) \]

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]]

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2211

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
)]

Rule 2204

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]

Rubi steps

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

Mathematica [A]  time = 0.0109012, 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 = 0.024, size = 44, normalized size = 0.9 \begin{align*}{f}^{a}x{f}^{{\frac{b}{{x}^{2}}}}-{{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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^a*x*f^(b/x^2)-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 = 1.23155, size = 35, normalized size = 0.71 \begin{align*} \frac{1}{2} \, f^{a} x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}} \Gamma \left (-\frac{1}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^a*x*sqrt(-b*log(f)/x^2)*gamma(-1/2, -b*log(f)/x^2)

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Fricas [A]  time = 1.78514, size = 104, normalized size = 2.12 \begin{align*} \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + f^{\frac{a x^{2} + b}{x^{2}}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}}\,{d x} \end{align*}

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)