\(\int (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}) \, dx\) [302]

Optimal result

Integrand size = 25, antiderivative size = 13 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\sqrt {2} x \sqrt {\log (x)} \]

[Out]

2^(1/2)*x*ln(x)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2336, 2211, 2235, 2333} \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\sqrt {2} x \sqrt {\log (x)} \]

[In]

Int[1/(Sqrt[2]*Sqrt[Log[x]]) + Sqrt[2]*Sqrt[Log[x]],x]

[Out]

Sqrt[2]*x*Sqrt[Log[x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

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 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\sqrt {2} x \sqrt {\log (x)} \]

[In]

Integrate[1/(Sqrt[2]*Sqrt[Log[x]]) + Sqrt[2]*Sqrt[Log[x]],x]

[Out]

Sqrt[2]*x*Sqrt[Log[x]]

Maple [F]

\[\int \left (\sqrt {2}\, \sqrt {\ln \left (x \right )}+\frac {\sqrt {2}}{2 \sqrt {\ln \left (x \right )}}\right )d x\]

[In]

int(2^(1/2)*ln(x)^(1/2)+1/2*2^(1/2)/ln(x)^(1/2),x)

[Out]

int(2^(1/2)*ln(x)^(1/2)+1/2*2^(1/2)/ln(x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(2^(1/2)*log(x)^(1/2)+1/2*2^(1/2)/log(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (12) = 24\).

Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 6.38 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } \sqrt {- \log {\left (x \right )}} \operatorname {erfc}{\left (\sqrt {- \log {\left (x \right )}} \right )}}{2 \sqrt {\log {\left (x \right )}}} + \frac {\sqrt {2} \left (x \sqrt {- \log {\left (x \right )}} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (x \right )}} \right )}}{2}\right ) \sqrt {\log {\left (x \right )}}}{\sqrt {- \log {\left (x \right )}}} \]

[In]

integrate(2**(1/2)*ln(x)**(1/2)+1/2*2**(1/2)/ln(x)**(1/2),x)

[Out]

sqrt(2)*sqrt(pi)*sqrt(-log(x))*erfc(sqrt(-log(x)))/(2*sqrt(log(x))) + sqrt(2)*(x*sqrt(-log(x)) + sqrt(pi)*erfc
(sqrt(-log(x)))/2)*sqrt(log(x))/sqrt(-log(x))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 3.15 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=-\frac {1}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (x\right )}\right ) - 2 \, x \sqrt {\log \left (x\right )}\right )} \]

[In]

integrate(2^(1/2)*log(x)^(1/2)+1/2*2^(1/2)/log(x)^(1/2),x, algorithm="maxima")

[Out]

-1/2*I*sqrt(2)*sqrt(pi)*erf(I*sqrt(log(x))) - 1/2*sqrt(2)*(-I*sqrt(pi)*erf(I*sqrt(log(x))) - 2*x*sqrt(log(x)))

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 3.15 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\frac {1}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {2} {\left (i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (x\right )}\right ) - 2 \, x \sqrt {\log \left (x\right )}\right )} \]

[In]

integrate(2^(1/2)*log(x)^(1/2)+1/2*2^(1/2)/log(x)^(1/2),x, algorithm="giac")

[Out]

1/2*I*sqrt(2)*sqrt(pi)*erf(-I*sqrt(log(x))) - 1/2*sqrt(2)*(I*sqrt(pi)*erf(-I*sqrt(log(x))) - 2*x*sqrt(log(x)))

Mupad [B] (verification not implemented)

Time = 16.43 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \left (\frac {1}{\sqrt {2} \sqrt {\log (x)}}+\sqrt {2} \sqrt {\log (x)}\right ) \, dx=\sqrt {2}\,x\,\sqrt {\ln \left (x\right )} \]

[In]

int(2^(1/2)/(2*log(x)^(1/2)) + 2^(1/2)*log(x)^(1/2),x)

[Out]

2^(1/2)*x*log(x)^(1/2)