∫ −10x−4e5x+x2+(−20x−8e5x+4x2) log(4ex)+(−10−4e5) log2(4ex)_______________________________________________

3.70.92 \(\int \frac {-10 x-4 e^5 x+x^2+(-20 x-8 e^5 x+4 x^2) \log (4 e^x)+(-10-4 e^5) \log ^2(4 e^x)}{2 x^4+8 x^3 \log ^2(4 e^x)+8 x^2 \log ^4(4 e^x)} \, dx\)

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

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Rubi [F] time = 1.34, 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 {-10 x-4 e^5 x+x^2+\left (-20 x-8 e^5 x+4 x^2\right ) \log \left (4 e^x\right )+\left (-10-4 e^5\right ) \log ^2\left (4 e^x\right )}{2 x^4+8 x^3 \log ^2\left (4 e^x\right )+8 x^2 \log ^4\left (4 e^x\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-10*x - 4*E^5*x + x^2 + (-20*x - 8*E^5*x + 4*x^2)*Log[4*E^x] + (-10 - 4*E^5)*Log[4*E^x]^2)/(2*x^4 + 8*x^3

*Log[4*E^x]^2 + 8*x^2*Log[4*E^x]^4),x]

[Out]

-1/2*1/(x + 2*Log[4*E^x]^2) - ((5 + 2*E^5)*Defer[Int][1/(x*(x + 2*Log[4*E^x]^2)^2), x])/2 - 2*(5 + 2*E^5)*Defe

r[Int][Log[4*E^x]/(x*(x + 2*Log[4*E^x]^2)^2), x] - ((5 + 2*E^5)*Defer[Int][1/(x^2*(x + 2*Log[4*E^x]^2)), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-10-4 e^5\right ) x+x^2+\left (-20 x-8 e^5 x+4 x^2\right ) \log \left (4 e^x\right )+\left (-10-4 e^5\right ) \log ^2\left (4 e^x\right )}{2 x^4+8 x^3 \log ^2\left (4 e^x\right )+8 x^2 \log ^4\left (4 e^x\right )} \, dx\\ &=\int \frac {\left (-10-4 e^5\right ) x+x^2+\left (-20 x-8 e^5 x+4 x^2\right ) \log \left (4 e^x\right )+\left (-10-4 e^5\right ) \log ^2\left (4 e^x\right )}{2 x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {\left (-10-4 e^5\right ) x+x^2+\left (-20 x-8 e^5 x+4 x^2\right ) \log \left (4 e^x\right )+\left (-10-4 e^5\right ) \log ^2\left (4 e^x\right )}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {\left (5+2 e^5-x\right ) \left (1+4 \log \left (4 e^x\right )\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2}+\frac {-5-2 e^5}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (5+2 e^5-x\right ) \left (1+4 \log \left (4 e^x\right )\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\right )+\frac {1}{2} \left (-5-2 e^5\right ) \int \frac {1}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {-1-4 \log \left (4 e^x\right )}{\left (x+2 \log ^2\left (4 e^x\right )\right )^2}+\frac {\left (5+2 e^5\right ) \left (1+4 \log \left (4 e^x\right )\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2}\right ) \, dx\right )+\frac {1}{2} \left (-5-2 e^5\right ) \int \frac {1}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {-1-4 \log \left (4 e^x\right )}{\left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\right )+\frac {1}{2} \left (-5-2 e^5\right ) \int \frac {1}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )} \, dx-\frac {1}{2} \left (5+2 e^5\right ) \int \frac {1+4 \log \left (4 e^x\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\\ &=-\frac {1}{2 \left (x+2 \log ^2\left (4 e^x\right )\right )}+\frac {1}{2} \left (-5-2 e^5\right ) \int \frac {1}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )} \, dx-\frac {1}{2} \left (5+2 e^5\right ) \int \left (\frac {1}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2}+\frac {4 \log \left (4 e^x\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2}\right ) \, dx\\ &=-\frac {1}{2 \left (x+2 \log ^2\left (4 e^x\right )\right )}+\frac {1}{2} \left (-5-2 e^5\right ) \int \frac {1}{x^2 \left (x+2 \log ^2\left (4 e^x\right )\right )} \, dx-\frac {1}{2} \left (5+2 e^5\right ) \int \frac {1}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx-\left (2 \left (5+2 e^5\right )\right ) \int \frac {\log \left (4 e^x\right )}{x \left (x+2 \log ^2\left (4 e^x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.38, size = 31, normalized size = 0.97 \begin {gather*} \frac {5+2 e^5-x}{2 x \left (x+2 \log ^2\left (4 e^x\right )\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-10*x - 4*E^5*x + x^2 + (-20*x - 8*E^5*x + 4*x^2)*Log[4*E^x] + (-10 - 4*E^5)*Log[4*E^x]^2)/(2*x^4 +

8*x^3*Log[4*E^x]^2 + 8*x^2*Log[4*E^x]^4),x]

[Out]

(5 + 2*E^5 - x)/(2*x*(x + 2*Log[4*E^x]^2))

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fricas [A] time = 0.59, size = 34, normalized size = 1.06 \begin {gather*} -\frac {x - 2 \, e^{5} - 5}{2 \, {\left (2 \, x^{3} + 8 \, x^{2} \log \relax (2) + 8 \, x \log \relax (2)^{2} + x^{2}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(5)-10)*log(4*exp(x))^2+(-8*x*exp(5)+4*x^2-20*x)*log(4*exp(x))-4*x*exp(5)+x^2-10*x)/(8*x^2*l

og(4*exp(x))^4+8*x^3*log(4*exp(x))^2+2*x^4),x, algorithm="fricas")

[Out]

-1/2*(x - 2*e^5 - 5)/(2*x^3 + 8*x^2*log(2) + 8*x*log(2)^2 + x^2)

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giac [A] time = 0.14, size = 34, normalized size = 1.06 \begin {gather*} -\frac {x - 2 \, e^{5} - 5}{2 \, {\left (2 \, x^{3} + 8 \, x^{2} \log \relax (2) + 8 \, x \log \relax (2)^{2} + x^{2}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(5)-10)*log(4*exp(x))^2+(-8*x*exp(5)+4*x^2-20*x)*log(4*exp(x))-4*x*exp(5)+x^2-10*x)/(8*x^2*l

og(4*exp(x))^4+8*x^3*log(4*exp(x))^2+2*x^4),x, algorithm="giac")

[Out]

-1/2*(x - 2*e^5 - 5)/(2*x^3 + 8*x^2*log(2) + 8*x*log(2)^2 + x^2)

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maple [A] time = 0.45, size = 40, normalized size = 1.25


method	result	size

risch	\(\frac {2 \,{\mathrm e}^{5}-x +5}{x \left (16 \ln \relax (2)^{2}+16 \ln \relax (2) \ln \left ({\mathrm e}^{x}\right )+4 \ln \left ({\mathrm e}^{x}\right )^{2}+2 x \right )}\)	\(40\)
default	\(-\frac {-2 \,{\mathrm e}^{5}-5}{4 \left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )^{2} x}+\frac {\left (-2 \,{\mathrm e}^{5}-5\right ) x -4 \,{\mathrm e}^{5} \left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )-\left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )^{2}-{\mathrm e}^{5}-\frac {5}{2}-10 \ln \left (4 \,{\mathrm e}^{x}\right )+10 x}{4 \left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )^{2} \left (x^{2}+2 x \left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )+\left (\ln \left (4 \,{\mathrm e}^{x}\right )-x \right )^{2}+\frac {x}{2}\right )}\)	\(119\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(5)-10)*ln(4*exp(x))^2+(-8*x*exp(5)+4*x^2-20*x)*ln(4*exp(x))-4*x*exp(5)+x^2-10*x)/(8*x^2*ln(4*exp(

x))^4+8*x^3*ln(4*exp(x))^2+2*x^4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B] time = 0.63, size = 5288, normalized size = 165.25 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(5)-10)*log(4*exp(x))^2+(-8*x*exp(5)+4*x^2-20*x)*log(4*exp(x))-4*x*exp(5)+x^2-10*x)/(8*x^2*l

og(4*exp(x))^4+8*x^3*log(4*exp(x))^2+2*x^4),x, algorithm="maxima")

[Out]

1/256*((512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2)

+ 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) + 16*(2*(16*lo

g(2)^2 + 16*log(2) + 1)*x^2 + 64*log(2)^3 + (64*log(2)^3 + 136*log(2)^2 + 24*log(2) + 1)*x + 4*log(2)^2)/(2*(1

6*log(2)^5 + log(2)^4)*x^3 + (128*log(2)^6 + 24*log(2)^5 + log(2)^4)*x^2 + 8*(16*log(2)^7 + log(2)^6)*x) - (8*

log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^6 + 2*(8*log(2) + 1)*log(x)/log(2)^6)*e^5*log(4*

e^x)^2 - 1/32*((256*log(2)^3 - 96*log(2)^2 - 24*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*

x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/((16*log(2)^5 + log(2)^4)*sqrt(16*log(2) + 1)) + 16*(2*x*(8*log(2) +

1) + 32*log(2)^2 + 16*log(2) + 1)/(128*log(2)^5 + 8*log(2)^4 + 2*(16*log(2)^3 + log(2)^2)*x^2 + (128*log(2)^4

+ 24*log(2)^3 + log(2)^2)*x) - log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^4 + 2*log(x)/log(2)^4)*e^5*lo

g(4*e^x) + 5/512*((512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1)

+ 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) + 1

6*(2*(16*log(2)^2 + 16*log(2) + 1)*x^2 + 64*log(2)^3 + (64*log(2)^3 + 136*log(2)^2 + 24*log(2) + 1)*x + 4*log(

2)^2)/(2*(16*log(2)^5 + log(2)^4)*x^3 + (128*log(2)^6 + 24*log(2)^5 + log(2)^4)*x^2 + 8*(16*log(2)^7 + log(2)^

6)*x) - (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^6 + 2*(8*log(2) + 1)*log(x)/log(2)^6)

*log(4*e^x)^2 - 1/2048*(4*(16*(32*log(2)^2 + 16*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/(16*log

(2)^5 + log(2)^4) - (512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(2) - 1)*((sqrt(16*log(2) + 1) + 8*log

(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x - sqrt(1

6*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1)

+ 8*log(2) + 1)))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) + 16*(8*log(2) + 1)*log((4*x - sqrt(16*log(2)

+ 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(sqrt(16*log(2) + 1)*log(2)^4) + 32*log(x)/l

og(2)^4 - (4*x*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*lo

g(2)^2) - sqrt(16*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*lo

g(2) + 1)) - 8*x)*(8*log(2) + 1)/log(2)^6 + 8*(x*log(x) - x)*(8*log(2) + 1)/log(2)^6)*log(4*e^x) - 16*(32*log(

2)^2 + 16*log(2) + 1)*(4*x*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2)

+ 1) + 8*log(2)^2) - sqrt(16*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2)

+ 1) + 8*log(2) + 1)) - 8*x)/(16*log(2)^5 + log(2)^4) - (512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(

2) - 1)*(8*x^2*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - ((

4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x - sqrt(16*log(2)

+ 1) - 8*log(2) - 1)*(sqrt(16*log(2) + 1) + 8*log(2) + 1) - ((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4

*x - sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x + sqrt(16*log(2) + 1) - 8*log(2) - 1)*(sqrt(16*log(2) + 1) - 8*

log(2) - 1) + (32*log(2)^2 + sqrt(16*log(2) + 1)*(8*log(2) + 1) + 16*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1)

+ 8*log(2) + 1) - (32*log(2)^2 - sqrt(16*log(2) + 1)*(8*log(2) + 1) + 16*log(2) + 1)*log(4*x - sqrt(16*log(2)

+ 1) + 8*log(2) + 1) - 4*x*sqrt(16*log(2) + 1))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) + 16*((sqrt(16

*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) -

1)*log(4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x +

sqrt(16*log(2) + 1) + 8*log(2) + 1)))*(8*log(2) + 1)/(sqrt(16*log(2) + 1)*log(2)^4) - 128*(x*log(x) - x)/log(2

)^4 + (8*x^2*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - 24*x^2 + (4*x*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2

)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - sqrt(16*log(2) + 1)*log((4*x - sqrt(16*log(

2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - 8*x)*(8*log(2) + 1) + 4*x*(8*log(2) + 1)

- (32*log(2)^2 + 16*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) + ((sqrt(16*log(2) + 1) + 8*log(2)

+ 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x - sqrt(16*l

og(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8

*log(2) + 1)))*sqrt(16*log(2) + 1) + (128*log(2)^2 + 24*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2)

+ 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/sqrt(16*log(2) + 1))*(8*log(2) + 1)/log(2)^6 - 8*(2*x^2*log(x

) - 3*x^2)*(8*log(2) + 1)/log(2)^6)*e^5 + 1/128*(32*(8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/

(16*log(2)^3 + log(2)^2) - (256*log(2)^3 - 96*log(2)^2 - 24*log(2) - 1)*((sqrt(16*log(2) + 1) + 8*log(2) + 1)*

log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x - sqrt(16*log(2)

+ 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2

) + 1)))/((16*log(2)^5 + log(2)^4)*sqrt(16*log(2) + 1)) + 32*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4

*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(sqrt(16*log(2) + 1)*log(2)^2) - (4*x*log(2*x^2 + x*(8*log(2) + 1) +

8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - sqrt(16*log(2) + 1)*log((4*x - sqrt

(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - 8*x)/log(2)^4 + 8*(x*log(x) - x)

/log(2)^4)*e^5 - 1/64*((256*log(2)^3 - 96*log(2)^2 - 24*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2)

+ 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/((16*log(2)^5 + log(2)^4)*sqrt(16*log(2) + 1)) + 16*(2*x*(8*l

og(2) + 1) + 32*log(2)^2 + 16*log(2) + 1)/(128*log(2)^5 + 8*log(2)^4 + 2*(16*log(2)^3 + log(2)^2)*x^2 + (128*l

og(2)^4 + 24*log(2)^3 + log(2)^2)*x) - log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^4 + 2*log(x)/log(2)^4

)*e^5 - 5/1024*(16*(32*log(2)^2 + 16*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/(16*log(2)^5 + log

(2)^4) - (512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(2) - 1)*((sqrt(16*log(2) + 1) + 8*log(2) + 1)*lo

g(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x - sqrt(16*log(2) +

1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2)

+ 1)))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) + 16*(8*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*l

og(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(sqrt(16*log(2) + 1)*log(2)^4) + 32*log(x)/log(2)^4 - (

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

qrt(16*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))

- 8*x)*(8*log(2) + 1)/log(2)^6 + 8*(x*log(x) - x)*(8*log(2) + 1)/log(2)^6)*log(4*e^x) - 5/64*((256*log(2)^3 -

96*log(2)^2 - 24*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log

(2) + 1))/((16*log(2)^5 + log(2)^4)*sqrt(16*log(2) + 1)) + 16*(2*x*(8*log(2) + 1) + 32*log(2)^2 + 16*log(2) +

1)/(128*log(2)^5 + 8*log(2)^4 + 2*(16*log(2)^3 + log(2)^2)*x^2 + (128*log(2)^4 + 24*log(2)^3 + log(2)^2)*x) -

log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^4 + 2*log(x)/log(2)^4)*log(4*e^x) - 2*((4*x + 8*log(2) + 1)/

(2*x^2*(16*log(2) + 1) + 128*log(2)^3 + (128*log(2)^2 + 24*log(2) + 1)*x + 8*log(2)^2) + 4*log((4*x - sqrt(16*

log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(16*log(2) + 1)^(3/2))*log(4*e^x) + 5/

256*(32*log(2)^2 + 16*log(2) + 1)*(4*x*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 +

x*(8*log(2) + 1) + 8*log(2)^2) - sqrt(16*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sq

rt(16*log(2) + 1) + 8*log(2) + 1)) - 8*x)/(16*log(2)^5 + log(2)^4) + 5/8*(8*log(2) + 1)*log(2*x^2 + x*(8*log(2

) + 1) + 8*log(2)^2)/(16*log(2)^3 + log(2)^2) + 5/4096*(512*log(2)^4 - 512*log(2)^3 - 288*log(2)^2 - 32*log(2)

- 1)*(8*x^2*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - ((4*

x + sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x - sqrt(16*log(2) +

1) - 8*log(2) - 1)*(sqrt(16*log(2) + 1) + 8*log(2) + 1) - ((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x

- sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x + sqrt(16*log(2) + 1) - 8*log(2) - 1)*(sqrt(16*log(2) + 1) - 8*lo

g(2) - 1) + (32*log(2)^2 + sqrt(16*log(2) + 1)*(8*log(2) + 1) + 16*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) +

8*log(2) + 1) - (32*log(2)^2 - sqrt(16*log(2) + 1)*(8*log(2) + 1) + 16*log(2) + 1)*log(4*x - sqrt(16*log(2) +

1) + 8*log(2) + 1) - 4*x*sqrt(16*log(2) + 1))/((16*log(2)^7 + log(2)^6)*sqrt(16*log(2) + 1)) - 5/256*(256*log

(2)^3 - 96*log(2)^2 - 24*log(2) - 1)*((sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*l

og(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x

- sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)))/((16*log(2)^5 + log(2)^4)*

sqrt(16*log(2) + 1)) - 5/128*(256*log(2)^3 - 96*log(2)^2 - 24*log(2) - 1)*log((4*x - sqrt(16*log(2) + 1) + 8*l

og(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/((16*log(2)^5 + log(2)^4)*sqrt(16*log(2) + 1)) - 5/8*(2

*x*(8*log(2) + 1) + 32*log(2)^2 + 16*log(2) + 1)/(128*log(2)^5 + 8*log(2)^4 + 2*(16*log(2)^3 + log(2)^2)*x^2 +

(128*log(2)^4 + 24*log(2)^3 + log(2)^2)*x) - 1/2*(4*x + 8*log(2) + 1)/(2*x^2*(16*log(2) + 1) + 128*log(2)^3 +

(128*log(2)^2 + 24*log(2) + 1)*x + 8*log(2)^2) + 2*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/(16*log(2) + 1)

- 2*((sqrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1

) - 8*log(2) - 1)*log(4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2

) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)))/(16*log(2) + 1)^(3/2) - 2*log((4*x - sqrt(16*log(2) + 1) +

8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(16*log(2) + 1)^(3/2) + 5/8*log((4*x - sqrt(16*log(

2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/(sqrt(16*log(2) + 1)*log(2)^2) - 5/256*((s

qrt(16*log(2) + 1) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*lo

g(2) - 1)*log(4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(

4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)))*(8*log(2) + 1)/(sqrt(16*log(2) + 1)*log(2)^4) - 5/256*(4*x*log(2*x

^2 + x*(8*log(2) + 1) + 8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - sqrt(16*log(

2) + 1)*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - 8*x)/log(

2)^4 + 5/16*(x*log(x) - x)/log(2)^4 + 5/128*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2)/log(2)^4 - 5/64*log(x)/

log(2)^4 - 5/4096*(8*x^2*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - 24*x^2 + (4*x*log(2*x^2 + x*(8*log(2) +

1) + 8*log(2)^2) + (8*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) - sqrt(16*log(2) + 1)*log((4*x -

sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1)) - 8*x)*(8*log(2) + 1) + 4*x*(8

*log(2) + 1) - (32*log(2)^2 + 16*log(2) + 1)*log(2*x^2 + x*(8*log(2) + 1) + 8*log(2)^2) + ((sqrt(16*log(2) + 1

) + 8*log(2) + 1)*log(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1) + (sqrt(16*log(2) + 1) - 8*log(2) - 1)*log(4*x

- sqrt(16*log(2) + 1) + 8*log(2) + 1) - 4*x*log((4*x - sqrt(16*log(2) + 1) + 8*log(2) + 1)/(4*x + sqrt(16*log

(2) + 1) + 8*log(2) + 1)))*sqrt(16*log(2) + 1) + (128*log(2)^2 + 24*log(2) + 1)*log((4*x - sqrt(16*log(2) + 1)

+ 8*log(2) + 1)/(4*x + sqrt(16*log(2) + 1) + 8*log(2) + 1))/sqrt(16*log(2) + 1))*(8*log(2) + 1)/log(2)^6 + 5/

512*(2*x^2*log(x) - 3*x^2)*(8*log(2) + 1)/log(2)^6

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mupad [B] time = 4.48, size = 35, normalized size = 1.09 \begin {gather*} \frac {2\,{\mathrm {e}}^5-x+5}{4\,x^3+\left (8\,\ln \relax (4)+2\right )\,x^2+4\,{\ln \relax (4)}^2\,x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x + 4*x*exp(5) + log(4*exp(x))^2*(4*exp(5) + 10) + log(4*exp(x))*(20*x + 8*x*exp(5) - 4*x^2) - x^2)/(

8*x^3*log(4*exp(x))^2 + 8*x^2*log(4*exp(x))^4 + 2*x^4),x)

[Out]

(2*exp(5) - x + 5)/(x^2*(8*log(4) + 2) + 4*x*log(4)^2 + 4*x^3)

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sympy [A] time = 8.87, size = 31, normalized size = 0.97 \begin {gather*} \frac {- x + 5 + 2 e^{5}}{4 x^{3} + x^{2} \left (2 + 16 \log {\relax (2 )}\right ) + 16 x \log {\relax (2 )}^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(5)-10)*ln(4*exp(x))**2+(-8*x*exp(5)+4*x**2-20*x)*ln(4*exp(x))-4*x*exp(5)+x**2-10*x)/(8*x**2

*ln(4*exp(x))**4+8*x**3*ln(4*exp(x))**2+2*x**4),x)

[Out]

(-x + 5 + 2*exp(5))/(4*x**3 + x**2*(2 + 16*log(2)) + 16*x*log(2)**2)

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