3.91.64 \(\int \frac {e^{\frac {-x+(-16-4 x) (i \pi -\log (\frac {5}{2}))+(4+x) \log (x)}{-4 (i \pi -\log (\frac {5}{2}))+\log (x)}} (-1-4 (i \pi -\log (\frac {5}{2}))-16 (i \pi -\log (\frac {5}{2}))^2+(1+8 (i \pi -\log (\frac {5}{2}))) \log (x)-\log ^2(x))}{16 (i \pi -\log (\frac {5}{2}))^2-8 (i \pi -\log (\frac {5}{2})) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ 3-e^{4+x+\frac {x}{4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-\log (x)}} \]

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Rubi [F]  time = 4.66, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I*Pi - Log[5/2]) + Log[x]))*(-1 - 4*(I*Pi -
Log[5/2]) - 16*(I*Pi - Log[5/2])^2 + (1 + 8*(I*Pi - Log[5/2]))*Log[x] - Log[x]^2))/(16*(I*Pi - Log[5/2])^2 -
8*(I*Pi - Log[5/2])*Log[x] + Log[x]^2),x]

[Out]

-Defer[Int][E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I*Pi - Log[5/2]) + Log[x])), x] + De
fer[Int][E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I*Pi - Log[5/2]) + Log[x]))/(4*Pi*(1 +
(I*Log[5/2])/Pi) + I*Log[x])^2, x] + I*Defer[Int][E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4
*(I*Pi - Log[5/2]) + Log[x]))/(4*Pi*(1 + (I*Log[5/2])/Pi) + I*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \left (1-4 \left (\pi +i \log \left (\frac {5}{2}\right )\right ) \left (-i+4 \pi +4 i \log \left (\frac {5}{2}\right )\right )-\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)+\log ^2(x)\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2} \, dx\\ &=\int \left (-\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )+\frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2}+\frac {i \exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)}\right ) \, dx\\ &=i \int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)} \, dx-\int \exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \, dx+\int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.36, size = 85, normalized size = 2.66 \begin {gather*} -e^{4+x+\frac {4 \pi (4+x)+i \left (16 \log \left (\frac {5}{2}\right )+x \left (-1+4 \log \left (\frac {5}{2}\right )\right )\right )}{4 \pi +4 i \log \left (\frac {5}{2}\right )+i \log (x)}} x^{(4+x) \left (-\frac {1}{\log (x)}+\frac {1}{-4 i \pi +4 \log \left (\frac {5}{2}\right )+\log (x)}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I*Pi - Log[5/2]) + Log[x]))*(-1 - 4*(
I*Pi - Log[5/2]) - 16*(I*Pi - Log[5/2])^2 + (1 + 8*(I*Pi - Log[5/2]))*Log[x] - Log[x]^2))/(16*(I*Pi - Log[5/2]
)^2 - 8*(I*Pi - Log[5/2])*Log[x] + Log[x]^2),x]

[Out]

-(E^(4 + x + (4*Pi*(4 + x) + I*(16*Log[5/2] + x*(-1 + 4*Log[5/2])))/(4*Pi + (4*I)*Log[5/2] + I*Log[x]))*x^((4
+ x)*(-Log[x]^(-1) + ((-4*I)*Pi + 4*Log[5/2] + Log[x])^(-1))))

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fricas [B]  time = 0.64, size = 115, normalized size = 3.59 \begin {gather*} -e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} - \frac {4 \, x \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} + \frac {x \log \relax (x)}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} - \frac {16 i \, \pi }{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} - \frac {x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} - \frac {16 \, \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)} + \frac {4 \, \log \relax (x)}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*log(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+
(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4*log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)
^2),x, algorithm="fricas")

[Out]

-e^(-4*I*pi*x/(-4*I*pi - 4*log(2/5) + log(x)) - 4*x*log(2/5)/(-4*I*pi - 4*log(2/5) + log(x)) + x*log(x)/(-4*I*
pi - 4*log(2/5) + log(x)) - 16*I*pi/(-4*I*pi - 4*log(2/5) + log(x)) - x/(-4*I*pi - 4*log(2/5) + log(x)) - 16*l
og(2/5)/(-4*I*pi - 4*log(2/5) + log(x)) + 4*log(x)/(-4*I*pi - 4*log(2/5) + log(x)))

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giac [B]  time = 2.32, size = 117, normalized size = 3.66 \begin {gather*} -e^{\left (\frac {4 \, \pi x}{4 \, \pi + 4 i \, \log \relax (5) - 4 i \, \log \relax (2) + i \, \log \relax (x)} + \frac {4 i \, x \log \relax (5)}{4 \, \pi + 4 i \, \log \relax (5) - 4 i \, \log \relax (2) + i \, \log \relax (x)} - \frac {4 i \, x \log \relax (2)}{4 \, \pi + 4 i \, \log \relax (5) - 4 i \, \log \relax (2) + i \, \log \relax (x)} + \frac {i \, x \log \relax (x)}{4 \, \pi + 4 i \, \log \relax (5) - 4 i \, \log \relax (2) + i \, \log \relax (x)} - \frac {i \, x}{4 \, \pi + 4 i \, \log \relax (5) - 4 i \, \log \relax (2) + i \, \log \relax (x)} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*log(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+
(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4*log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)
^2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 2.85, size = 62, normalized size = 1.94

method	result	size
risch	\(-{\mathrm e}^{\frac {4 i \pi x -x \ln \relax (x )+16 i \pi -4 x \ln \relax (5)+4 x \ln \relax (2)-4 \ln \relax (x )-16 \ln \relax (5)+16 \ln \relax (2)+x}{-4 \ln \relax (5)+4 \ln \relax (2)+4 i \pi -\ln \relax (x )}}\)	\(62\)
norman	\(\frac {\left (-16 \pi ^{2}-16 \ln \relax (5)^{2}+32 \ln \relax (2) \ln \relax (5)-16 \ln \relax (2)^{2}\right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \relax (x )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \relax (x )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}+\left (-8 \ln \relax (5)+8 \ln \relax (2)\right ) \ln \relax (x ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \relax (x )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \relax (x )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}-\ln \relax (x )^{2} {\mathrm e}^{\frac {\left (4+x \right ) \ln \relax (x )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \relax (x )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}}{16 \pi ^{2}+16 \ln \relax (5)^{2}-32 \ln \relax (2) \ln \relax (5)+8 \ln \relax (5) \ln \relax (x )+16 \ln \relax (2)^{2}-8 \ln \relax (2) \ln \relax (x )+\ln \relax (x )^{2}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)^2+(8*ln(2/5)+8*I*Pi+1)*ln(x)-16*(ln(2/5)+I*Pi)^2-4*ln(2/5)-4*I*Pi-1)*exp(((4+x)*ln(x)+(-16-4*x)*(l
n(2/5)+I*Pi)-x)/(ln(x)-4*ln(2/5)-4*I*Pi))/(ln(x)^2-8*(ln(2/5)+I*Pi)*ln(x)+16*(ln(2/5)+I*Pi)^2),x,method=_RETUR
NVERBOSE)

[Out]

-exp((4*I*Pi*x-x*ln(x)+16*I*Pi-4*x*ln(5)+4*x*ln(2)-4*ln(x)-16*ln(5)+16*ln(2)+x)/(-4*ln(5)+4*ln(2)+4*I*Pi-ln(x)
))

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maxima [B]  time = 3.33, size = 184, normalized size = 5.75 \begin {gather*} -e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} + \frac {4 \, x \log \relax (5)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} - \frac {4 \, x \log \relax (2)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} + \frac {x \log \relax (x)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} - \frac {16 i \, \pi }{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} - \frac {x}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} + \frac {16 \, \log \relax (5)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} - \frac {16 \, \log \relax (2)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)} + \frac {4 \, \log \relax (x)}{-4 i \, \pi + 4 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*log(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+
(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4*log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)
^2),x, algorithm="maxima")

[Out]

-e^(-4*I*pi*x/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + 4*x*log(5)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) -
4*x*log(2)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + x*log(x)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) - 16*
I*pi/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) - x/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + 16*log(5)/(-4*I*p
i + 4*log(5) - 4*log(2) + log(x)) - 16*log(2)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + 4*log(x)/(-4*I*pi + 4
*log(5) - 4*log(2) + log(x)))

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mupad [B]  time = 10.28, size = 82, normalized size = 2.56 \begin {gather*} -{\mathrm {e}}^{\frac {\Pi \,16{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {x}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {\Pi \,x\,4{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}}\,{\left (\frac {625\,x}{16}\right )}^{\frac {x\,1{}\mathrm {i}+4{}\mathrm {i}}{4\,\Pi -\ln \left (\frac {2}{5}\right )\,4{}\mathrm {i}+\ln \relax (x)\,1{}\mathrm {i}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x + (4*x + 16)*(Pi*1i + log(2/5)) - log(x)*(x + 4))/(Pi*4i + 4*log(2/5) - log(x)))*(Pi*4i + 4*log(2
/5) - log(x)*(Pi*8i + 8*log(2/5) + 1) + log(x)^2 + 16*(Pi*1i + log(2/5))^2 + 1))/(log(x)^2 - 8*log(x)*(Pi*1i +
log(2/5)) + 16*(Pi*1i + log(2/5))^2),x)

[Out]

-exp((Pi*16i)/(Pi*4i - log((625*x)/16)) + x/(Pi*4i - log((625*x)/16)) + (Pi*x*4i)/(Pi*4i - log((625*x)/16)))*(
(625*x)/16)^((x*1i + 4i)/(4*Pi - log(2/5)*4i + log(x)*1i))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)**2+(8*ln(2/5)+8*I*pi+1)*ln(x)-16*(ln(2/5)+I*pi)**2-4*ln(2/5)-4*I*pi-1)*exp(((4+x)*ln(x)+(-16
-4*x)*(ln(2/5)+I*pi)-x)/(ln(x)-4*ln(2/5)-4*I*pi))/(ln(x)**2-8*(ln(2/5)+I*pi)*ln(x)+16*(ln(2/5)+I*pi)**2),x)

[Out]

Timed out

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