3.49.54 \(\int \frac {1}{2} (6 e^{6 x^2} x+(i \pi +\log (5))^2) \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{4} \left (e^{6 x^2}+2 x (i \pi +\log (5))^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 2209} \begin {gather*} \frac {e^{6 x^2}}{4}+\frac {1}{2} x (\log (5)+i \pi )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6*E^(6*x^2)*x + (I*Pi + Log[5])^2)/2,x]

[Out]

E^(6*x^2)/4 + (x*(I*Pi + Log[5])^2)/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (6 e^{6 x^2} x+(i \pi +\log (5))^2\right ) \, dx\\ &=\frac {1}{2} x (i \pi +\log (5))^2+3 \int e^{6 x^2} x \, dx\\ &=\frac {e^{6 x^2}}{4}+\frac {1}{2} x (i \pi +\log (5))^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (\frac {e^{6 x^2}}{2}+x (i \pi +\log (5))^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*E^(6*x^2)*x + (I*Pi + Log[5])^2)/2,x]

[Out]

(E^(6*x^2)/2 + x*(I*Pi + Log[5])^2)/2

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fricas [A]  time = 0.67, size = 28, normalized size = 1.12 \begin {gather*} -\frac {1}{2} \, \pi ^{2} x + i \, \pi x \log \relax (5) + \frac {1}{2} \, x \log \relax (5)^{2} + \frac {1}{4} \, e^{\left (6 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x*exp(6*x^2)+1/2*(log(5)+I*pi)^2,x, algorithm="fricas")

[Out]

-1/2*pi^2*x + I*pi*x*log(5) + 1/2*x*log(5)^2 + 1/4*e^(6*x^2)

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giac [A]  time = 0.24, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, {\left (i \, \pi + \log \relax (5)\right )}^{2} x + \frac {1}{4} \, e^{\left (6 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x*exp(6*x^2)+1/2*(log(5)+I*pi)^2,x, algorithm="giac")

[Out]

1/2*(I*pi + log(5))^2*x + 1/4*e^(6*x^2)

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maple [A]  time = 0.04, size = 30, normalized size = 1.20

method	result	size
default	\(-\frac {x \,\pi ^{2}}{2}+i \pi \ln \relax (5) x +\frac {x \ln \relax (5)^{2}}{2}+\frac {{\mathrm e}^{6 x^{2}}}{4}\)	\(30\)
norman	\(\left (i \pi \ln \relax (5)-\frac {\pi ^{2}}{2}+\frac {\ln \relax (5)^{2}}{2}\right ) x +\frac {{\mathrm e}^{6 x^{2}}}{4}\)	\(30\)
risch	\(-\frac {x \,\pi ^{2}}{2}+i \pi \ln \relax (5) x +\frac {x \ln \relax (5)^{2}}{2}+\frac {{\mathrm e}^{6 x^{2}}}{4}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x*exp(6*x^2)+1/2*(ln(5)+I*Pi)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x*Pi^2+I*Pi*ln(5)*x+1/2*x*ln(5)^2+1/4*exp(6*x^2)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, {\left (i \, \pi + \log \relax (5)\right )}^{2} x + \frac {1}{4} \, e^{\left (6 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x*exp(6*x^2)+1/2*(log(5)+I*pi)^2,x, algorithm="maxima")

[Out]

1/2*(I*pi + log(5))^2*x + 1/4*e^(6*x^2)

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mupad [B]  time = 0.12, size = 21, normalized size = 0.84 \begin {gather*} \frac {{\mathrm {e}}^{6\,x^2}}{4}+\frac {x\,{\left (\ln \relax (5)+\Pi \,1{}\mathrm {i}\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x*exp(6*x^2) + (Pi*1i + log(5))^2/2,x)

[Out]

exp(6*x^2)/4 + (x*(Pi*1i + log(5))^2)/2

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sympy [A]  time = 0.14, size = 27, normalized size = 1.08 \begin {gather*} x \left (- \frac {\pi ^{2}}{2} + \frac {\log {\relax (5 )}^{2}}{2} + i \pi \log {\relax (5 )}\right ) + \frac {e^{6 x^{2}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x*exp(6*x**2)+1/2*(ln(5)+I*pi)**2,x)

[Out]

x*(-pi**2/2 + log(5)**2/2 + I*pi*log(5)) + exp(6*x**2)/4

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