3.14.59 \(\int \frac {4^{-x} (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2})^x (1-2 \log (x)+(4 x^2+x^2 \log (7)+\log (x)) \log (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}))}{4 x^2+x^2 \log (7)+\log (x)} \, dx\)

Optimal. Leaf size=18 \[ 4^{-x} \left (4+\log (7)+\frac {\log (x)}{x^2}\right )^x \]

________________________________________________________________________________________

Rubi [F]  time = 3.98, 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 {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(((4*x^2 + x^2*Log[7] + Log[x])/x^2)^x*(1 - 2*Log[x] + (4*x^2 + x^2*Log[7] + Log[x])*Log[(4*x^2 + x^2*Log[
7] + Log[x])/(4*x^2)]))/(4^x*(4*x^2 + x^2*Log[7] + Log[x])),x]

[Out]

-Defer[Int][2^(1 - 2*x)*(4*(1 + Log[7]/4) + Log[x]/x^2)^x, x] + Defer[Int][(4*(1 + Log[7]/4) + Log[x]/x^2)^x/(
4^x*(x^2*(4 + Log[7]) + Log[x])), x] + (4 + Log[7])*Defer[Int][(2^(1 - 2*x)*x^2*(4*(1 + Log[7]/4) + Log[x]/x^2
)^x)/(x^2*(4 + Log[7]) + Log[x]), x] + Defer[Int][((4*(1 + Log[7]/4) + Log[x]/x^2)^x*Log[(4*(1 + Log[7]/4) + L
og[x]/x^2)/4])/4^x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{x^2 (4+\log (7))+\log (x)} \, dx\\ &=\int \frac {4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{x^2 (4+\log (7))+\log (x)} \, dx\\ &=\int \left (\frac {4^{-x} (1-2 \log (x)) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)}+4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right )\right ) \, dx\\ &=\int \frac {4^{-x} (1-2 \log (x)) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=\int \frac {4^{-x} (1-2 \log (x)) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{x^2 (4+\log (7))+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=\int \left (-2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x+\frac {4^{-x} \left (1+2 x^2 (4+\log (7))\right ) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)}\right ) \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=-\int 2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \, dx+\int \frac {4^{-x} \left (1+2 x^2 (4+\log (7))\right ) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=-\int 2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \, dx+\int \frac {4^{-x} \left (1+2 x^2 (4+\log (7))\right ) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{x^2 (4+\log (7))+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=-\int 2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \, dx+\int \left (\frac {4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)}+\frac {2^{1-2 x} x^2 (4+\log (7)) \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)}\right ) \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=(4+\log (7)) \int \frac {2^{1-2 x} x^2 \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)} \, dx-\int 2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \, dx+\int \frac {4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ &=(4+\log (7)) \int \frac {2^{1-2 x} x^2 \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{x^2 (4+\log (7))+\log (x)} \, dx-\int 2^{1-2 x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \, dx+\int \frac {4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x}{x^2 (4+\log (7))+\log (x)} \, dx+\int 4^{-x} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )^x \log \left (\frac {1}{4} \left (4 \left (1+\frac {\log (7)}{4}\right )+\frac {\log (x)}{x^2}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.13, size = 18, normalized size = 1.00 \begin {gather*} 4^{-x} \left (4+\log (7)+\frac {\log (x)}{x^2}\right )^x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(((4*x^2 + x^2*Log[7] + Log[x])/x^2)^x*(1 - 2*Log[x] + (4*x^2 + x^2*Log[7] + Log[x])*Log[(4*x^2 + x^
2*Log[7] + Log[x])/(4*x^2)]))/(4^x*(4*x^2 + x^2*Log[7] + Log[x])),x]

[Out]

(4 + Log[7] + Log[x]/x^2)^x/4^x

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 21, normalized size = 1.17 \begin {gather*} \left (\frac {x^{2} \log \relax (7) + 4 \, x^{2} + \log \relax (x)}{4 \, x^{2}}\right )^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2)-2*log(x)+1)*exp(x*log(1/4*(log(x)+
x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*log(7)+4*x^2),x, algorithm="fricas")

[Out]

(1/4*(x^2*log(7) + 4*x^2 + log(x))/x^2)^x

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{2} \log \relax (7) + 4 \, x^{2} + \log \relax (x)\right )} \log \left (\frac {x^{2} \log \relax (7) + 4 \, x^{2} + \log \relax (x)}{4 \, x^{2}}\right ) - 2 \, \log \relax (x) + 1\right )} \left (\frac {x^{2} \log \relax (7) + 4 \, x^{2} + \log \relax (x)}{4 \, x^{2}}\right )^{x}}{x^{2} \log \relax (7) + 4 \, x^{2} + \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2)-2*log(x)+1)*exp(x*log(1/4*(log(x)+
x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*log(7)+4*x^2),x, algorithm="giac")

[Out]

integrate(((x^2*log(7) + 4*x^2 + log(x))*log(1/4*(x^2*log(7) + 4*x^2 + log(x))/x^2) - 2*log(x) + 1)*(1/4*(x^2*
log(7) + 4*x^2 + log(x))/x^2)^x/(x^2*log(7) + 4*x^2 + log(x)), x)

________________________________________________________________________________________

maple [C]  time = 0.15, size = 236, normalized size = 13.11

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )}{x^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )\right )+4 \ln \relax (x )+4 \ln \relax (2)-2 \ln \left (\ln \relax (x )+x^{2} \ln \relax (7)+4 x^{2}\right )\right )}{2}}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)+x^2*ln(7)+4*x^2)*ln(1/4*(ln(x)+x^2*ln(7)+4*x^2)/x^2)-2*ln(x)+1)*exp(x*ln(1/4*(ln(x)+x^2*ln(7)+4*x^
2)/x^2))/(ln(x)+x^2*ln(7)+4*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(-1/2*x*(-I*Pi*csgn(I*x^2)^3+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x^2)*csgn(I*x)^2+I*Pi*csgn(I*(ln(x)
+x^2*ln(7)+4*x^2)/x^2)^3-I*Pi*csgn(I*(ln(x)+x^2*ln(7)+4*x^2)/x^2)^2*csgn(I/x^2)-I*Pi*csgn(I*(ln(x)+x^2*ln(7)+4
*x^2)/x^2)^2*csgn(I*(ln(x)+x^2*ln(7)+4*x^2))+I*Pi*csgn(I*(ln(x)+x^2*ln(7)+4*x^2)/x^2)*csgn(I/x^2)*csgn(I*(ln(x
)+x^2*ln(7)+4*x^2))+4*ln(x)+4*ln(2)-2*ln(ln(x)+x^2*ln(7)+4*x^2)))

________________________________________________________________________________________

maxima [A]  time = 0.95, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (-2 \, x \log \relax (2) + x \log \left (x^{2} {\left (\log \relax (7) + 4\right )} + \log \relax (x)\right ) - 2 \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2)-2*log(x)+1)*exp(x*log(1/4*(log(x)+
x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*log(7)+4*x^2),x, algorithm="maxima")

[Out]

e^(-2*x*log(2) + x*log(x^2*(log(7) + 4) + log(x)) - 2*x*log(x))

________________________________________________________________________________________

mupad [B]  time = 1.18, size = 29, normalized size = 1.61 \begin {gather*} \frac {{\left (\frac {1}{x^2}\right )}^x\,{\left (\ln \relax (x)+x^2\,\ln \relax (7)+4\,x^2\right )}^x}{2^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*log((log(x)/4 + (x^2*log(7))/4 + x^2)/x^2))*(log((log(x)/4 + (x^2*log(7))/4 + x^2)/x^2)*(log(x) + x
^2*log(7) + 4*x^2) - 2*log(x) + 1))/(log(x) + x^2*log(7) + 4*x^2),x)

[Out]

((1/x^2)^x*(log(x) + x^2*log(7) + 4*x^2)^x)/2^(2*x)

________________________________________________________________________________________

sympy [A]  time = 0.98, size = 24, normalized size = 1.33 \begin {gather*} e^{x \log {\left (\frac {\frac {x^{2} \log {\relax (7 )}}{4} + x^{2} + \frac {\log {\relax (x )}}{4}}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(x)+x**2*ln(7)+4*x**2)*ln(1/4*(ln(x)+x**2*ln(7)+4*x**2)/x**2)-2*ln(x)+1)*exp(x*ln(1/4*(ln(x)+x**
2*ln(7)+4*x**2)/x**2))/(ln(x)+x**2*ln(7)+4*x**2),x)

[Out]

exp(x*log((x**2*log(7)/4 + x**2 + log(x)/4)/x**2))

________________________________________________________________________________________