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

Optimal. Leaf size=21 \[ -\frac {6}{-1-\frac {e^8}{4}+e^x+x-\log (4)} \]

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Rubi [A]  time = 0.21, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6688, 12, 6686} \begin {gather*} \frac {24}{-4 x-4 e^x+e^8+4+\log (256)} \end {gather*}

Antiderivative was successfully verified.

[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 6686

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.90 \begin {gather*} \frac {24}{4+e^8-4 e^x-4 x+\log (256)} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.70, size = 21, normalized size = 1.00 \begin {gather*} -\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \relax (2) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.14, size = 21, normalized size = 1.00 \begin {gather*} -\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \relax (2) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.16, size = 20, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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)-4*exp(x)+8*ln(2)-4*x+4)

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maxima [A]  time = 0.49, size = 21, normalized size = 1.00 \begin {gather*} -\frac {24}{4 \, x - e^{8} + 4 \, e^{x} - 8 \, \log \relax (2) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 7.56, size = 35, normalized size = 1.67 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.15, size = 20, normalized size = 0.95 \begin {gather*} - \frac {24}{4 x + 4 e^{x} - e^{8} - 8 \log {\relax (2 )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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