3.81.26 \(\int \frac {e^x (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} (-32 x+16 x^2)+e^{10} (192 x-32 x^3)-256 \log (2))}{6400+1440 x^2+960 x^3+241 x^4+e^{40} x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} (-12 x^4-4 x^5)+e^{20} (160 x^2+54 x^4+36 x^5+6 x^6)+e^{10} (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7)+(-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} (192 x^2+64 x^3)) \log (2)+256 \log ^2(2)} \, dx\) [8026]

Optimal. Leaf size=29 \[ \frac {e^x}{5+\frac {1}{16} x^2 \left (3-e^{10}+x\right )^2-\log (2)} \]

[Out]

exp(x)/(1/16*x^2*(x-exp(10)+3)^2+5-ln(2))

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Rubi [A]
time = 2.33, antiderivative size = 44, normalized size of antiderivative = 1.52, number of steps used = 4, number of rules used = 4, integrand size = 222, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6, 6820, 12, 2327} \begin {gather*} \frac {16 e^x}{x^4+2 \left (3-e^{10}\right ) x^3+\left (3-e^{10}\right )^2 x^2+16 (5-\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(1280 - 288*x - 144*x^2 + 32*x^3 + 16*x^4 + E^20*(-32*x + 16*x^2) + E^10*(192*x - 32*x^3) - 256*Log[2
]))/(6400 + 1440*x^2 + 960*x^3 + 241*x^4 + E^40*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + E^30*(-12*x^4 - 4*x^5)
+ E^20*(160*x^2 + 54*x^4 + 36*x^5 + 6*x^6) + E^10*(-960*x^2 - 320*x^3 - 108*x^4 - 108*x^5 - 36*x^6 - 4*x^7) +
(-2560 - 288*x^2 - 32*E^20*x^2 - 192*x^3 - 32*x^4 + E^10*(192*x^2 + 64*x^3))*Log[2] + 256*Log[2]^2),x]

[Out]

(16*E^x)/((3 - E^10)^2*x^2 + 2*(3 - E^10)*x^3 + x^4 + 16*(5 - Log[2]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 2327

Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]/Coefficient[z, x, Exponent[z, x]])*F^u*v^(n + 1), x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]

Rule 6820

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 {e^x \left (1280-288 x-144 x^2+32 x^3+16 x^4+e^{20} \left (-32 x+16 x^2\right )+e^{10} \left (192 x-32 x^3\right )-256 \log (2)\right )}{6400+1440 x^2+960 x^3+\left (241+e^{40}\right ) x^4+108 x^5+54 x^6+12 x^7+x^8+e^{30} \left (-12 x^4-4 x^5\right )+e^{20} \left (160 x^2+54 x^4+36 x^5+6 x^6\right )+e^{10} \left (-960 x^2-320 x^3-108 x^4-108 x^5-36 x^6-4 x^7\right )+\left (-2560-288 x^2-32 e^{20} x^2-192 x^3-32 x^4+e^{10} \left (192 x^2+64 x^3\right )\right ) \log (2)+256 \log ^2(2)} \, dx\\ &=\int \frac {16 e^x \left (-2 \left (3-e^{10}\right )^2 x-\left (9-e^{20}\right ) x^2+2 \left (1-e^{10}\right ) x^3+x^4+16 (5-\log (2))\right )}{\left (\left (3-e^{10}\right )^2 x^2+2 \left (3-e^{10}\right ) x^3+x^4+16 (5-\log (2))\right )^2} \, dx\\ &=16 \int \frac {e^x \left (-2 \left (3-e^{10}\right )^2 x-\left (9-e^{20}\right ) x^2+2 \left (1-e^{10}\right ) x^3+x^4+16 (5-\log (2))\right )}{\left (\left (3-e^{10}\right )^2 x^2+2 \left (3-e^{10}\right ) x^3+x^4+16 (5-\log (2))\right )^2} \, dx\\ &=\frac {16 e^x}{\left (3-e^{10}\right )^2 x^2+2 \left (3-e^{10}\right ) x^3+x^4+16 (5-\log (2))}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.30, size = 38, normalized size = 1.31 \begin {gather*} \frac {16 e^x}{\left (-3+e^{10}\right )^2 x^2-2 \left (-3+e^{10}\right ) x^3+x^4-16 (-5+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1280 - 288*x - 144*x^2 + 32*x^3 + 16*x^4 + E^20*(-32*x + 16*x^2) + E^10*(192*x - 32*x^3) - 256
*Log[2]))/(6400 + 1440*x^2 + 960*x^3 + 241*x^4 + E^40*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + E^30*(-12*x^4 -
4*x^5) + E^20*(160*x^2 + 54*x^4 + 36*x^5 + 6*x^6) + E^10*(-960*x^2 - 320*x^3 - 108*x^4 - 108*x^5 - 36*x^6 - 4*
x^7) + (-2560 - 288*x^2 - 32*E^20*x^2 - 192*x^3 - 32*x^4 + E^10*(192*x^2 + 64*x^3))*Log[2] + 256*Log[2]^2),x]

[Out]

(16*E^x)/((-3 + E^10)^2*x^2 - 2*(-3 + E^10)*x^3 + x^4 - 16*(-5 + Log[2]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.77, size = 10159, normalized size = 350.31

method	result	size
gosper	\(-\frac {16 \,{\mathrm e}^{x}}{-x^{2} {\mathrm e}^{20}+2 x^{3} {\mathrm e}^{10}-x^{4}+6 \,{\mathrm e}^{10} x^{2}-6 x^{3}-9 x^{2}+16 \ln \left (2\right )-80}\)	\(51\)
norman	\(-\frac {16 \,{\mathrm e}^{x}}{-x^{2} {\mathrm e}^{20}+2 x^{3} {\mathrm e}^{10}-x^{4}+6 \,{\mathrm e}^{10} x^{2}-6 x^{3}-9 x^{2}+16 \ln \left (2\right )-80}\)	\(51\)
default	\(\text {Expression too large to display}\)	\(10159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-256*ln(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16*x^4+32*x^3-144*x^2-288*x+1280)*exp(x)/(256*
ln(2)^2+(-32*x^2*exp(10)^2+(64*x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*ln(2)+x^4*exp(10)^4+(-4*x^5-1
2*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36*x^6-108*x^5-108*x^4-320*x^3-960*x^2)*exp(1
0)+x^8+12*x^7+54*x^6+108*x^5+241*x^4+960*x^3+1440*x^2+6400),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.58, size = 36, normalized size = 1.24 \begin {gather*} \frac {16 \, e^{x}}{x^{4} - 2 \, x^{3} {\left (e^{10} - 3\right )} + x^{2} {\left (e^{20} - 6 \, e^{10} + 9\right )} - 16 \, \log \left (2\right ) + 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16*x^4+32*x^3-144*x^2-288*x+1280)*exp(x
)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64*x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+
(-4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36*x^6-108*x^5-108*x^4-320*x^3-960*x
^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+241*x^4+960*x^3+1440*x^2+6400),x, algorithm="maxima")

[Out]

16*e^x/(x^4 - 2*x^3*(e^10 - 3) + x^2*(e^20 - 6*e^10 + 9) - 16*log(2) + 80)

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Fricas [A]
time = 0.43, size = 44, normalized size = 1.52 \begin {gather*} \frac {16 \, e^{x}}{x^{4} + 6 \, x^{3} + x^{2} e^{20} + 9 \, x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{10} - 16 \, \log \left (2\right ) + 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16*x^4+32*x^3-144*x^2-288*x+1280)*exp(x
)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64*x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+
(-4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36*x^6-108*x^5-108*x^4-320*x^3-960*x
^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+241*x^4+960*x^3+1440*x^2+6400),x, algorithm="fricas")

[Out]

16*e^x/(x^4 + 6*x^3 + x^2*e^20 + 9*x^2 - 2*(x^3 + 3*x^2)*e^10 - 16*log(2) + 80)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
time = 0.20, size = 48, normalized size = 1.66 \begin {gather*} \frac {16 e^{x}}{x^{4} - 2 x^{3} e^{10} + 6 x^{3} - 6 x^{2} e^{10} + 9 x^{2} + x^{2} e^{20} - 16 \log {\left (2 \right )} + 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*ln(2)+(16*x**2-32*x)*exp(10)**2+(-32*x**3+192*x)*exp(10)+16*x**4+32*x**3-144*x**2-288*x+1280)*
exp(x)/(256*ln(2)**2+(-32*x**2*exp(10)**2+(64*x**3+192*x**2)*exp(10)-32*x**4-192*x**3-288*x**2-2560)*ln(2)+x**
4*exp(10)**4+(-4*x**5-12*x**4)*exp(10)**3+(6*x**6+36*x**5+54*x**4+160*x**2)*exp(10)**2+(-4*x**7-36*x**6-108*x*
*5-108*x**4-320*x**3-960*x**2)*exp(10)+x**8+12*x**7+54*x**6+108*x**5+241*x**4+960*x**3+1440*x**2+6400),x)

[Out]

16*exp(x)/(x**4 - 2*x**3*exp(10) + 6*x**3 - 6*x**2*exp(10) + 9*x**2 + x**2*exp(20) - 16*log(2) + 80)

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Giac [A]
time = 1.23, size = 45, normalized size = 1.55 \begin {gather*} \frac {32 \, e^{x}}{x^{4} - 2 \, x^{3} e^{10} + 6 \, x^{3} + x^{2} e^{20} - 6 \, x^{2} e^{10} + 9 \, x^{2} - 16 \, \log \left (2\right ) + 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*log(2)+(16*x^2-32*x)*exp(10)^2+(-32*x^3+192*x)*exp(10)+16*x^4+32*x^3-144*x^2-288*x+1280)*exp(x
)/(256*log(2)^2+(-32*x^2*exp(10)^2+(64*x^3+192*x^2)*exp(10)-32*x^4-192*x^3-288*x^2-2560)*log(2)+x^4*exp(10)^4+
(-4*x^5-12*x^4)*exp(10)^3+(6*x^6+36*x^5+54*x^4+160*x^2)*exp(10)^2+(-4*x^7-36*x^6-108*x^5-108*x^4-320*x^3-960*x
^2)*exp(10)+x^8+12*x^7+54*x^6+108*x^5+241*x^4+960*x^3+1440*x^2+6400),x, algorithm="giac")

[Out]

32*e^x/(x^4 - 2*x^3*e^10 + 6*x^3 + x^2*e^20 - 6*x^2*e^10 + 9*x^2 - 16*log(2) + 80)

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Mupad [B]
time = 6.32, size = 36, normalized size = 1.24 \begin {gather*} \frac {16\,{\mathrm {e}}^x}{x^4+\left (6-2\,{\mathrm {e}}^{10}\right )\,x^3+{\left ({\mathrm {e}}^{10}-3\right )}^2\,x^2-16\,\ln \left (2\right )+80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(288*x + 256*log(2) + exp(20)*(32*x - 16*x^2) - exp(10)*(192*x - 32*x^3) + 144*x^2 - 32*x^3 - 16*
x^4 - 1280))/(x^4*exp(40) - exp(30)*(12*x^4 + 4*x^5) - exp(10)*(960*x^2 + 320*x^3 + 108*x^4 + 108*x^5 + 36*x^6
+ 4*x^7) + 256*log(2)^2 + 1440*x^2 + 960*x^3 + 241*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 - log(2)*(32*x^2*exp
(20) - exp(10)*(192*x^2 + 64*x^3) + 288*x^2 + 192*x^3 + 32*x^4 + 2560) + exp(20)*(160*x^2 + 54*x^4 + 36*x^5 +
6*x^6) + 6400),x)

[Out]

(16*exp(x))/(x^2*(exp(10) - 3)^2 - x^3*(2*exp(10) - 6) - 16*log(2) + x^4 + 80)

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