\(\int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x (-32-8 e^8+32 x-32 \log (4))+(32+8 e^8-32 x) \log (4)+16 \log ^2(4)} \, dx\) [8083]

Optimal result

Integrand size = 76, antiderivative size = 21 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {6}{-1-\frac {e^8}{4}+e^x+x-\log (4)} \]

[Out]

-6/(exp(x)-2*ln(2)-1-1/4*exp(4)^2+x)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6820, 12, 6818} \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {24}{-4 x-4 e^x+e^8+4+\log (256)} \]

[In]

Int[(96 + 96*E^x)/(16 + E^16 + 16*E^(2*x) + E^8*(8 - 8*x) - 32*x + 16*x^2 + E^x*(-32 - 8*E^8 + 32*x - 32*Log[4
]) + (32 + 8*E^8 - 32*x)*Log[4] + 16*Log[4]^2),x]

[Out]

24/(4 + E^8 - 4*E^x - 4*x + Log[256])

Rule 12

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

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {96 \left (1+e^x\right )}{\left (4 e^x+4 x-4 \left (1+\frac {e^8}{4}+\log (4)\right )\right )^2} \, dx \\ & = 96 \int \frac {1+e^x}{\left (4 e^x+4 x-4 \left (1+\frac {e^8}{4}+\log (4)\right )\right )^2} \, dx \\ & = \frac {24}{4+e^8-4 e^x-4 x+\log (256)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {24}{4+e^8-4 e^x-4 x+\log (256)} \]

[In]

Integrate[(96 + 96*E^x)/(16 + E^16 + 16*E^(2*x) + E^8*(8 - 8*x) - 32*x + 16*x^2 + E^x*(-32 - 8*E^8 + 32*x - 32
*Log[4]) + (32 + 8*E^8 - 32*x)*Log[4] + 16*Log[4]^2),x]

[Out]

24/(4 + E^8 - 4*E^x - 4*x + Log[256])

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\)	\(20\)
norman	\(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\)	\(22\)
parallelrisch	\(\frac {24}{{\mathrm e}^{8}+8 \ln \left (2\right )-4 \,{\mathrm e}^{x}-4 x +4}\)	\(22\)

[In]

int((96*exp(x)+96)/(16*exp(x)^2+(-64*ln(2)-8*exp(4)^2+32*x-32)*exp(x)+64*ln(2)^2+2*(8*exp(4)^2-32*x+32)*ln(2)+
exp(4)^4+(-8*x+8)*exp(4)^2+16*x^2-32*x+16),x,method=_RETURNVERBOSE)

[Out]

24/(exp(8)+8*ln(2)-4*exp(x)-4*x+4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]

[In]

integrate((96*exp(x)+96)/(16*exp(x)^2+(-64*log(2)-8*exp(4)^2+32*x-32)*exp(x)+64*log(2)^2+2*(8*exp(4)^2-32*x+32
)*log(2)+exp(4)^4+(-8*x+8)*exp(4)^2+16*x^2-32*x+16),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=- \frac {6}{x + e^{x} - \frac {e^{8}}{4} - 2 \log {\left (2 \right )} - 1} \]

[In]

integrate((96*exp(x)+96)/(16*exp(x)**2+(-64*ln(2)-8*exp(4)**2+32*x-32)*exp(x)+64*ln(2)**2+2*(8*exp(4)**2-32*x+
32)*ln(2)+exp(4)**4+(-8*x+8)*exp(4)**2+16*x**2-32*x+16),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]

[In]

integrate((96*exp(x)+96)/(16*exp(x)^2+(-64*log(2)-8*exp(4)^2+32*x-32)*exp(x)+64*log(2)^2+2*(8*exp(4)^2-32*x+32
)*log(2)+exp(4)^4+(-8*x+8)*exp(4)^2+16*x^2-32*x+16),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=-\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \left (2\right ) - 4} \]

[In]

integrate((96*exp(x)+96)/(16*exp(x)^2+(-64*log(2)-8*exp(4)^2+32*x-32)*exp(x)+64*log(2)^2+2*(8*exp(4)^2-32*x+32
)*log(2)+exp(4)^4+(-8*x+8)*exp(4)^2+16*x^2-32*x+16),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {96+96 e^x}{16+e^{16}+16 e^{2 x}+e^8 (8-8 x)-32 x+16 x^2+e^x \left (-32-8 e^8+32 x-32 \log (4)\right )+\left (32+8 e^8-32 x\right ) \log (4)+16 \log ^2(4)} \, dx=\frac {96\,\left (\frac {{\mathrm {e}}^8}{4}+\frac {\ln \left (256\right )}{4}+1\right )}{\left ({\mathrm {e}}^8+\ln \left (256\right )+4\right )\,\left ({\mathrm {e}}^8-4\,x+\ln \left (256\right )-4\,{\mathrm {e}}^x+4\right )} \]

[In]

int((96*exp(x) + 96)/(16*exp(2*x) - 32*x + exp(16) - exp(x)*(8*exp(8) - 32*x + 64*log(2) + 32) + 2*log(2)*(8*e
xp(8) - 32*x + 32) + 64*log(2)^2 + 16*x^2 - exp(8)*(8*x - 8) + 16),x)

[Out]

(96*(exp(8)/4 + log(256)/4 + 1))/((exp(8) + log(256) + 4)*(exp(8) - 4*x + log(256) - 4*exp(x) + 4))