3.11.5 \(\int \frac {1}{3} (-3-64 e^{\frac {4}{3} (x+(i \pi +\log (2))^2)}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} (x+(i \pi +\log (2))^2)} \log ^2(4)+64 e^{\frac {1}{3} (x+(i \pi +\log (2))^2)} \log ^3(4)) \, dx\)

Optimal. Leaf size=32 \[ 4-x-16 \left (-e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )}+\log (4)\right )^4 \]

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Rubi [B]  time = 0.06, antiderivative size = 90, normalized size of antiderivative = 2.81, number of steps used = 6, number of rules used = 2, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 2194} \begin {gather*} -x+64 \log ^3(4) e^{\frac {1}{3} \left (x+(\log (2)+i \pi )^2\right )}-96 \log ^2(4) e^{\frac {2}{3} \left (x+(\log (2)+i \pi )^2\right )}-16 e^{\frac {4}{3} \left (x+(\log (2)+i \pi )^2\right )}+64 \log (4) e^{x+(\log (2)+i \pi )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 64*E^((4*(x + (I*Pi + Log[2])^2))/3) + 192*E^(x + (I*Pi + Log[2])^2)*Log[4] - 192*E^((2*(x + (I*Pi +
Log[2])^2))/3)*Log[4]^2 + 64*E^((x + (I*Pi + Log[2])^2)/3)*Log[4]^3)/3,x]

[Out]

-16*E^((4*(x + (I*Pi + Log[2])^2))/3) - x + 64*E^(x + (I*Pi + Log[2])^2)*Log[4] - 96*E^((2*(x + (I*Pi + Log[2]
)^2))/3)*Log[4]^2 + 64*E^((x + (I*Pi + Log[2])^2)/3)*Log[4]^3

Rule 12

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx\\ &=-x-\frac {64}{3} \int e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx+(64 \log (4)) \int e^{x+(i \pi +\log (2))^2} \, dx-\left (64 \log ^2(4)\right ) \int e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx+\frac {1}{3} \left (64 \log ^3(4)\right ) \int e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx\\ &=-16 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}-x+64 e^{x+(i \pi +\log (2))^2} \log (4)-96 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.14, size = 148, normalized size = 4.62 \begin {gather*} \frac {1}{3} \left (-48 e^{\frac {4 x}{3}+\frac {4}{3} \left (-\pi ^2+2 i \pi \log (2)+\log ^2(2)\right )}-3 x+192 e^{x+(i \pi +\log (2))^2} \log (4)-288 e^{\frac {2 x}{3}+\frac {2}{3} \left (-\pi ^2+2 i \pi \log (2)+\log ^2(2)\right )} \log ^2(4)+192 e^{\frac {x}{3}+\frac {1}{3} \left (\pi ^2-2 i \pi \log (2)-\log ^2(2)\right )+\frac {2}{3} \left (-\pi ^2+2 i \pi \log (2)+\log ^2(2)\right )} \log ^3(4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 64*E^((4*(x + (I*Pi + Log[2])^2))/3) + 192*E^(x + (I*Pi + Log[2])^2)*Log[4] - 192*E^((2*(x + (
I*Pi + Log[2])^2))/3)*Log[4]^2 + 64*E^((x + (I*Pi + Log[2])^2)/3)*Log[4]^3)/3,x]

[Out]

(-48*E^((4*x)/3 + (4*(-Pi^2 + (2*I)*Pi*Log[2] + Log[2]^2))/3) - 3*x + 192*E^(x + (I*Pi + Log[2])^2)*Log[4] - 2
88*E^((2*x)/3 + (2*(-Pi^2 + (2*I)*Pi*Log[2] + Log[2]^2))/3)*Log[4]^2 + 192*E^(x/3 + (Pi^2 - (2*I)*Pi*Log[2] -
Log[2]^2)/3 + (2*(-Pi^2 + (2*I)*Pi*Log[2] + Log[2]^2))/3)*Log[4]^3)/3

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fricas [B]  time = 0.76, size = 102, normalized size = 3.19 \begin {gather*} 512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \relax (2) + \frac {1}{3} \, \log \relax (2)^{2} + \frac {1}{3} \, x\right )} \log \relax (2)^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \relax (2) + \frac {2}{3} \, \log \relax (2)^{2} + \frac {2}{3} \, x\right )} \log \relax (2)^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \relax (2) + \log \relax (2)^{2} + x\right )} \log \relax (2) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \relax (2) + \frac {4}{3} \, \log \relax (2)^{2} + \frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-64/3*exp(1/3*(log(2)+I*pi)^2+1/3*x)^4+128*log(2)*exp(1/3*(log(2)+I*pi)^2+1/3*x)^3-256*log(2)^2*exp(
1/3*(log(2)+I*pi)^2+1/3*x)^2+512/3*log(2)^3*exp(1/3*(log(2)+I*pi)^2+1/3*x)-1,x, algorithm="fricas")

[Out]

512*e^(-1/3*pi^2 + 2/3*I*pi*log(2) + 1/3*log(2)^2 + 1/3*x)*log(2)^3 - 384*e^(-2/3*pi^2 + 4/3*I*pi*log(2) + 2/3
*log(2)^2 + 2/3*x)*log(2)^2 + 128*e^(-pi^2 + 2*I*pi*log(2) + log(2)^2 + x)*log(2) - x - 16*e^(-4/3*pi^2 + 8/3*
I*pi*log(2) + 4/3*log(2)^2 + 4/3*x)

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giac [B]  time = 0.37, size = 102, normalized size = 3.19 \begin {gather*} 512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \relax (2) + \frac {1}{3} \, \log \relax (2)^{2} + \frac {1}{3} \, x\right )} \log \relax (2)^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \relax (2) + \frac {2}{3} \, \log \relax (2)^{2} + \frac {2}{3} \, x\right )} \log \relax (2)^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \relax (2) + \log \relax (2)^{2} + x\right )} \log \relax (2) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \relax (2) + \frac {4}{3} \, \log \relax (2)^{2} + \frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-64/3*exp(1/3*(log(2)+I*pi)^2+1/3*x)^4+128*log(2)*exp(1/3*(log(2)+I*pi)^2+1/3*x)^3-256*log(2)^2*exp(
1/3*(log(2)+I*pi)^2+1/3*x)^2+512/3*log(2)^3*exp(1/3*(log(2)+I*pi)^2+1/3*x)-1,x, algorithm="giac")

[Out]

512*e^(-1/3*pi^2 + 2/3*I*pi*log(2) + 1/3*log(2)^2 + 1/3*x)*log(2)^3 - 384*e^(-2/3*pi^2 + 4/3*I*pi*log(2) + 2/3
*log(2)^2 + 2/3*x)*log(2)^2 + 128*e^(-pi^2 + 2*I*pi*log(2) + log(2)^2 + x)*log(2) - x - 16*e^(-4/3*pi^2 + 8/3*
I*pi*log(2) + 4/3*log(2)^2 + 4/3*x)

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maple [B]  time = 0.18, size = 93, normalized size = 2.91

method	result	size
default	\(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \relax (2) {\mathrm e}^{\left (\ln \relax (2)+i \pi \right )^{2}+x}-384 \ln \relax (2)^{2} {\mathrm e}^{\frac {2 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \relax (2)^{3} {\mathrm e}^{\frac {\left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {x}{3}}\)	\(93\)
norman	\(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \relax (2) {\mathrm e}^{\left (\ln \relax (2)+i \pi \right )^{2}+x}-384 \ln \relax (2)^{2} {\mathrm e}^{\frac {2 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \relax (2)^{3} {\mathrm e}^{\frac {\left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {x}{3}}\)	\(93\)
derivativedivides	\(-16 \,{\mathrm e}^{\frac {4 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \relax (2) {\mathrm e}^{\left (\ln \relax (2)+i \pi \right )^{2}+x}-384 \ln \relax (2)^{2} {\mathrm e}^{\frac {2 \left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \relax (2)^{3} {\mathrm e}^{\frac {\left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {x}{3}}-3 \ln \left ({\mathrm e}^{\frac {\left (\ln \relax (2)+i \pi \right )^{2}}{3}+\frac {x}{3}}\right )\)	\(109\)
risch	\(-x -16 \,2^{\frac {8 i \pi }{3}} {\mathrm e}^{-\frac {4 \pi ^{2}}{3}+\frac {4 \ln \relax (2)^{2}}{3}+\frac {4 x}{3}}+128 \ln \relax (2) 2^{2 i \pi } {\mathrm e}^{-\pi ^{2}+\ln \relax (2)^{2}+x}-384 \ln \relax (2)^{2} 2^{\frac {4 i \pi }{3}} {\mathrm e}^{-\frac {2 \pi ^{2}}{3}+\frac {2 \ln \relax (2)^{2}}{3}+\frac {2 x}{3}}+512 \ln \relax (2)^{3} 2^{\frac {2 i \pi }{3}} {\mathrm e}^{-\frac {\pi ^{2}}{3}+\frac {\ln \relax (2)^{2}}{3}+\frac {x}{3}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-64/3*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)^4+128*ln(2)*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)^3-256*ln(2)^2*exp(1/3*(ln(2)
+I*Pi)^2+1/3*x)^2+512/3*ln(2)^3*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)-1,x,method=_RETURNVERBOSE)

[Out]

-x-16*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)^4+128*ln(2)*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)^3-384*ln(2)^2*exp(1/3*(ln(2)+I*P
i)^2+1/3*x)^2+512*ln(2)^3*exp(1/3*(ln(2)+I*Pi)^2+1/3*x)

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maxima [B]  time = 0.54, size = 102, normalized size = 3.19 \begin {gather*} 512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \relax (2) + \frac {1}{3} \, \log \relax (2)^{2} + \frac {1}{3} \, x\right )} \log \relax (2)^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \relax (2) + \frac {2}{3} \, \log \relax (2)^{2} + \frac {2}{3} \, x\right )} \log \relax (2)^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \relax (2) + \log \relax (2)^{2} + x\right )} \log \relax (2) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \relax (2) + \frac {4}{3} \, \log \relax (2)^{2} + \frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-64/3*exp(1/3*(log(2)+I*pi)^2+1/3*x)^4+128*log(2)*exp(1/3*(log(2)+I*pi)^2+1/3*x)^3-256*log(2)^2*exp(
1/3*(log(2)+I*pi)^2+1/3*x)^2+512/3*log(2)^3*exp(1/3*(log(2)+I*pi)^2+1/3*x)-1,x, algorithm="maxima")

[Out]

512*e^(-1/3*pi^2 + 2/3*I*pi*log(2) + 1/3*log(2)^2 + 1/3*x)*log(2)^3 - 384*e^(-2/3*pi^2 + 4/3*I*pi*log(2) + 2/3
*log(2)^2 + 2/3*x)*log(2)^2 + 128*e^(-pi^2 + 2*I*pi*log(2) + log(2)^2 + x)*log(2) - x - 16*e^(-4/3*pi^2 + 8/3*
I*pi*log(2) + 4/3*log(2)^2 + 4/3*x)

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mupad [B]  time = 0.37, size = 106, normalized size = 3.31 \begin {gather*} -x-16\,2^{\frac {\Pi \,8{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {4\,\Pi ^2}{3}+\frac {4\,x}{3}+\frac {4\,{\ln \relax (2)}^2}{3}}+512\,2^{\frac {\Pi \,2{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {\Pi ^2}{3}+\frac {x}{3}+\frac {{\ln \relax (2)}^2}{3}}\,{\ln \relax (2)}^3-384\,2^{\frac {\Pi \,4{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {2\,\Pi ^2}{3}+\frac {2\,x}{3}+\frac {2\,{\ln \relax (2)}^2}{3}}\,{\ln \relax (2)}^2+128\,2^{\Pi \,2{}\mathrm {i}}\,{\mathrm {e}}^{-\Pi ^2+x+{\ln \relax (2)}^2}\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((512*exp(x/3 + (Pi*1i + log(2))^2/3)*log(2)^3)/3 - (64*exp((4*x)/3 + (4*(Pi*1i + log(2))^2)/3))/3 - 256*ex
p((2*x)/3 + (2*(Pi*1i + log(2))^2)/3)*log(2)^2 + 128*exp(x + (Pi*1i + log(2))^2)*log(2) - 1,x)

[Out]

512*2^((Pi*2i)/3)*exp(x/3 - Pi^2/3 + log(2)^2/3)*log(2)^3 - 16*2^((Pi*8i)/3)*exp((4*x)/3 - (4*Pi^2)/3 + (4*log
(2)^2)/3) - x - 384*2^((Pi*4i)/3)*exp((2*x)/3 - (2*Pi^2)/3 + (2*log(2)^2)/3)*log(2)^2 + 128*2^(Pi*2i)*exp(x -
Pi^2 + log(2)^2)*log(2)

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sympy [B]  time = 0.32, size = 163, normalized size = 5.09 \begin {gather*} - x + \frac {- 16 e^{2 \pi ^{2}} e^{\frac {4 x}{3}} e^{\frac {8 i \pi \log {\relax (2 )}}{3}} e^{\frac {4 \log {\relax (2 )}^{2}}{3}} - 384 e^{\frac {8 \pi ^{2}}{3}} e^{\frac {2 x}{3}} e^{\frac {4 i \pi \log {\relax (2 )}}{3}} e^{\frac {2 \log {\relax (2 )}^{2}}{3}} \log {\relax (2 )}^{2} + 512 e^{3 \pi ^{2}} e^{\frac {x}{3}} e^{\frac {2 i \pi \log {\relax (2 )}}{3}} e^{\frac {\log {\relax (2 )}^{2}}{3}} \log {\relax (2 )}^{3} + 128 e^{\frac {7 \pi ^{2}}{3}} e^{x} e^{2 i \pi \log {\relax (2 )}} e^{\log {\relax (2 )}^{2}} \log {\relax (2 )}}{e^{\frac {10 \pi ^{2}}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-64/3*exp(1/3*(ln(2)+I*pi)**2+1/3*x)**4+128*ln(2)*exp(1/3*(ln(2)+I*pi)**2+1/3*x)**3-256*ln(2)**2*exp
(1/3*(ln(2)+I*pi)**2+1/3*x)**2+512/3*ln(2)**3*exp(1/3*(ln(2)+I*pi)**2+1/3*x)-1,x)

[Out]

-x + (-16*exp(2*pi**2)*exp(4*x/3)*exp(8*I*pi*log(2)/3)*exp(4*log(2)**2/3) - 384*exp(8*pi**2/3)*exp(2*x/3)*exp(
4*I*pi*log(2)/3)*exp(2*log(2)**2/3)*log(2)**2 + 512*exp(3*pi**2)*exp(x/3)*exp(2*I*pi*log(2)/3)*exp(log(2)**2/3
)*log(2)**3 + 128*exp(7*pi**2/3)*exp(x)*exp(2*I*pi*log(2))*exp(log(2)**2)*log(2))*exp(-10*pi**2/3)

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