3.116 \(\int x \sqrt {\log (a x^n)} \, dx\)

Optimal. Leaf size=72 \[ \frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {n} x^2 \left (a x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \]

[Out]

-1/8*x^2*erfi(2^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*n^(1/2)*2^(1/2)*Pi^(1/2)/((a*x^n)^(2/n))+1/2*x^2*ln(a*x^n)^(1/2
)

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Rubi [A]  time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2305, 2310, 2180, 2204} \[ \frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {n} x^2 \left (a x^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[Log[a*x^n]],x]

[Out]

-(Sqrt[n]*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(4*(a*x^n)^(2/n)) + (x^2*Sqrt[Log[a*x^n]])/
2

Rule 2180

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] &&  !$UseGamma === True

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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2310

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

Rubi steps

\begin {align*} \int x \sqrt {\log \left (a x^n\right )} \, dx &=\frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}-\frac {1}{4} n \int \frac {x}{\sqrt {\log \left (a x^n\right )}} \, dx\\ &=\frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}-\frac {1}{4} \left (x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )\\ &=\frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}-\frac {1}{2} \left (x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int e^{\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )\\ &=-\frac {1}{4} \sqrt {n} \sqrt {\frac {\pi }{2}} x^2 \left (a x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{2} x^2 \sqrt {\log \left (a x^n\right )}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 0.93 \[ \frac {1}{8} x^2 \left (4 \sqrt {\log \left (a x^n\right )}-\sqrt {2 \pi } \sqrt {n} \left (a x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[Log[a*x^n]],x]

[Out]

(x^2*(-((Sqrt[n]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(2/n)) + 4*Sqrt[Log[a*x^n]]))/8

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\log \left (a x^{n}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*sqrt(log(a*x^n)), x)

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int x \sqrt {\ln \left (a \,x^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(a*x^n)^(1/2),x)

[Out]

int(x*ln(a*x^n)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\log \left (a x^{n}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*sqrt(log(a*x^n)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {\ln \left (a\,x^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(a*x^n)^(1/2),x)

[Out]

int(x*log(a*x^n)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\log {\left (a x^{n} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(a*x**n)**(1/2),x)

[Out]

Integral(x*sqrt(log(a*x**n)), x)

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