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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 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]

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
)]

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*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.00 \[ x f^{a+\frac {b}{x^2}}-\sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \]

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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fricas [A]  time = 0.41, size = 42, normalized size = 0.86 \[ \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) + f^{\frac {a x^{2} + b}{x^{2}}} x \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 44, normalized size = 0.90 \[ -\frac {\sqrt {\pi }\, b \,f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right ) \ln \relax (f )}{\sqrt {-b \ln \relax (f )}}+x \,f^{a} f^{\frac {b}{x^{2}}} \]

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 = 1.25, size = 26, normalized size = 0.53 \[ \frac {1}{2} \, f^{a} x \sqrt {-\frac {b \log \relax (f)}{x^{2}}} \Gamma \left (-\frac {1}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]

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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mupad [B]  time = 3.60, size = 44, normalized size = 0.90 \[ f^a\,f^{\frac {b}{x^2}}\,x-\frac {b\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (f)}{x\,\sqrt {b\,\ln \relax (f)}}\right )\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{2}}}\, dx \]

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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