3.98.22 \(\int (2 x+2 (i \pi +\log (4))^2 \log (4 e^x)) \, dx\)

Optimal. Leaf size=25 \[ 3+e+x^2+(i \pi +\log (4))^2 \log ^2\left (4 e^x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2157, 30} \begin {gather*} x^2+(\log (4)+i \pi )^2 \log ^2\left (4 e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + 2*(I*Pi + Log[4])^2*Log[4*E^x],x]

[Out]

x^2 + (I*Pi + Log[4])^2*Log[4*E^x]^2

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.28 \begin {gather*} 2 \left (\frac {x^2}{2}+\frac {1}{2} (i \pi +\log (4))^2 \log ^2\left (4 e^x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + 2*(I*Pi + Log[4])^2*Log[4*E^x],x]

[Out]

2*(x^2/2 + ((I*Pi + Log[4])^2*Log[4*E^x]^2)/2)

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fricas [A]  time = 1.02, size = 50, normalized size = 2.00 \begin {gather*} 16 \, x \log \relax (2)^{3} - {\left (\pi ^{2} - 1\right )} x^{2} - 4 \, {\left (-4 i \, \pi x - x^{2}\right )} \log \relax (2)^{2} - 4 \, {\left (\pi ^{2} x - i \, \pi x^{2}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(2*log(2)+I*pi)^2*log(4*exp(x))+2*x,x, algorithm="fricas")

[Out]

16*x*log(2)^3 - (pi^2 - 1)*x^2 - 4*(-4*I*pi*x - x^2)*log(2)^2 - 4*(pi^2*x - I*pi*x^2)*log(2)

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giac [A]  time = 0.25, size = 24, normalized size = 0.96 \begin {gather*} {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{2} {\left (x^{2} + 4 \, x \log \relax (2)\right )} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(2*log(2)+I*pi)^2*log(4*exp(x))+2*x,x, algorithm="giac")

[Out]

(I*pi + 2*log(2))^2*(x^2 + 4*x*log(2)) + x^2

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maple [A]  time = 0.05, size = 37, normalized size = 1.48

method	result	size
norman	\(\left (4 i \pi \ln \relax (2)-\pi ^{2}+4 \ln \relax (2)^{2}-1\right ) \ln \left (4 \,{\mathrm e}^{x}\right )^{2}+2 x \ln \left (4 \,{\mathrm e}^{x}\right )\)	\(37\)
default	\(x^{2}-\pi ^{2} \ln \left (4 \,{\mathrm e}^{x}\right )^{2}+4 i \pi \ln \relax (2) \ln \left (4 \,{\mathrm e}^{x}\right )^{2}+4 \ln \relax (2)^{2} \ln \left (4 \,{\mathrm e}^{x}\right )^{2}\)	\(43\)
risch	\(2 \left (2 \ln \relax (2)+i \pi \right )^{2} x \ln \left ({\mathrm e}^{x}\right )-4 \pi ^{2} \ln \relax (2) x +16 i \pi \ln \relax (2)^{2} x +16 x \ln \relax (2)^{3}+\pi ^{2} x^{2}-4 i \pi \ln \relax (2) x^{2}-4 x^{2} \ln \relax (2)^{2}+x^{2}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*(2*ln(2)+I*Pi)^2*ln(4*exp(x))+2*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 22, normalized size = 0.88 \begin {gather*} {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{2} \log \left (4 \, e^{x}\right )^{2} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(2*log(2)+I*pi)^2*log(4*exp(x))+2*x,x, algorithm="maxima")

[Out]

(I*pi + 2*log(2))^2*log(4*e^x)^2 + x^2

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mupad [B]  time = 5.96, size = 50, normalized size = 2.00 \begin {gather*} \left (-\Pi ^2+4{}\mathrm {i}\,\ln \relax (2)\,\Pi +4\,{\ln \relax (2)}^2+1\right )\,x^2+\left (-2\,\ln \relax (4)\,\Pi ^2+8{}\mathrm {i}\,\ln \relax (2)\,\ln \relax (4)\,\Pi +8\,{\ln \relax (2)}^2\,\ln \relax (4)\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 2*log(4*exp(x))*(Pi*1i + 2*log(2))^2,x)

[Out]

x*(8*log(2)^2*log(4) - 2*Pi^2*log(4) + Pi*log(2)*log(4)*8i) + x^2*(Pi*log(2)*4i - Pi^2 + 4*log(2)^2 + 1)

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sympy [A]  time = 0.11, size = 49, normalized size = 1.96 \begin {gather*} x^{2} \left (- \pi ^{2} + 1 + 4 \log {\relax (2 )}^{2} + 4 i \pi \log {\relax (2 )}\right ) + x \left (- 4 \pi ^{2} \log {\relax (2 )} + 16 \log {\relax (2 )}^{3} + 16 i \pi \log {\relax (2 )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*(2*ln(2)+I*pi)**2*ln(4*exp(x))+2*x,x)

[Out]

x**2*(-pi**2 + 1 + 4*log(2)**2 + 4*I*pi*log(2)) + x*(-4*pi**2*log(2) + 16*log(2)**3 + 16*I*pi*log(2)**2)

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