3.32.18 \(\int \frac {-16 x^7+7 x^8+e^x (20 x^4-16 x^5+x^6+x^7)+(8 x^7+e^x (-20 x^4+6 x^5+2 x^6)) \log (3+\log (2))+e^x (5 x^4+x^5) \log ^2(3+\log (2))}{4-4 x+x^2+(-4+2 x) \log (3+\log (2))+\log ^2(3+\log (2))} \, dx\)

Optimal. Leaf size=22 \[ x^5 \left (e^x+\frac {x^3}{-2+x+\log (3+\log (2))}\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 16, number of rules used = 5, integrand size = 113, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6688, 2196, 2176, 2194, 74} \begin {gather*} e^x x^5-\frac {x^8}{-x+2-\log (3+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*x^7 + 7*x^8 + E^x*(20*x^4 - 16*x^5 + x^6 + x^7) + (8*x^7 + E^x*(-20*x^4 + 6*x^5 + 2*x^6))*Log[3 + Log
[2]] + E^x*(5*x^4 + x^5)*Log[3 + Log[2]]^2)/(4 - 4*x + x^2 + (-4 + 2*x)*Log[3 + Log[2]] + Log[3 + Log[2]]^2),x
]

[Out]

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

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 \left (e^x x^4 (5+x)+\frac {x^7 (7 x+8 (-2+\log (3+\log (2))))}{(-2+x+\log (3+\log (2)))^2}\right ) \, dx\\ &=\int e^x x^4 (5+x) \, dx+\int \frac {x^7 (7 x+8 (-2+\log (3+\log (2))))}{(-2+x+\log (3+\log (2)))^2} \, dx\\ &=-\frac {x^8}{2-x-\log (3+\log (2))}+\int \left (5 e^x x^4+e^x x^5\right ) \, dx\\ &=-\frac {x^8}{2-x-\log (3+\log (2))}+5 \int e^x x^4 \, dx+\int e^x x^5 \, dx\\ &=5 e^x x^4+e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}-5 \int e^x x^4 \, dx-20 \int e^x x^3 \, dx\\ &=-20 e^x x^3+e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}+20 \int e^x x^3 \, dx+60 \int e^x x^2 \, dx\\ &=60 e^x x^2+e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}-60 \int e^x x^2 \, dx-120 \int e^x x \, dx\\ &=-120 e^x x+e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}+120 \int e^x \, dx+120 \int e^x x \, dx\\ &=120 e^x+e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}-120 \int e^x \, dx\\ &=e^x x^5-\frac {x^8}{2-x-\log (3+\log (2))}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 56, normalized size = 2.55 \begin {gather*} \frac {e^x x^6+x^8+e^x x^5 (-2+\log (3+\log (2)))+x (-2+\log (3+\log (2)))^7+(-2+\log (3+\log (2)))^8}{-2+x+\log (3+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^7 + 7*x^8 + E^x*(20*x^4 - 16*x^5 + x^6 + x^7) + (8*x^7 + E^x*(-20*x^4 + 6*x^5 + 2*x^6))*Log[3
+ Log[2]] + E^x*(5*x^4 + x^5)*Log[3 + Log[2]]^2)/(4 - 4*x + x^2 + (-4 + 2*x)*Log[3 + Log[2]] + Log[3 + Log[2]
]^2),x]

[Out]

(E^x*x^6 + x^8 + E^x*x^5*(-2 + Log[3 + Log[2]]) + x*(-2 + Log[3 + Log[2]])^7 + (-2 + Log[3 + Log[2]])^8)/(-2 +
x + Log[3 + Log[2]])

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fricas [B]  time = 1.34, size = 132, normalized size = 6.00 \begin {gather*} \frac {x^{8} + {\left (x - 16\right )} \log \left (\log \relax (2) + 3\right )^{7} + \log \left (\log \relax (2) + 3\right )^{8} - 14 \, {\left (x - 8\right )} \log \left (\log \relax (2) + 3\right )^{6} + 28 \, {\left (3 \, x - 16\right )} \log \left (\log \relax (2) + 3\right )^{5} - 280 \, {\left (x - 4\right )} \log \left (\log \relax (2) + 3\right )^{4} + 112 \, {\left (5 \, x - 16\right )} \log \left (\log \relax (2) + 3\right )^{3} - 224 \, {\left (3 \, x - 8\right )} \log \left (\log \relax (2) + 3\right )^{2} + {\left (x^{6} - 2 \, x^{5}\right )} e^{x} + {\left (x^{5} e^{x} + 448 \, x - 1024\right )} \log \left (\log \relax (2) + 3\right ) - 128 \, x + 256}{x + \log \left (\log \relax (2) + 3\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x)+8*x^7)*log(3+log(2))+(x^7+x^6-16*x^
5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm="fricas")

[Out]

(x^8 + (x - 16)*log(log(2) + 3)^7 + log(log(2) + 3)^8 - 14*(x - 8)*log(log(2) + 3)^6 + 28*(3*x - 16)*log(log(2
) + 3)^5 - 280*(x - 4)*log(log(2) + 3)^4 + 112*(5*x - 16)*log(log(2) + 3)^3 - 224*(3*x - 8)*log(log(2) + 3)^2
+ (x^6 - 2*x^5)*e^x + (x^5*e^x + 448*x - 1024)*log(log(2) + 3) - 128*x + 256)/(x + log(log(2) + 3) - 2)

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giac [B]  time = 0.23, size = 178, normalized size = 8.09 \begin {gather*} \frac {x^{8} + x \log \left (\log \relax (2) + 3\right )^{7} + \log \left (\log \relax (2) + 3\right )^{8} + x^{6} e^{x} + x^{5} e^{x} \log \left (\log \relax (2) + 3\right ) - 14 \, x \log \left (\log \relax (2) + 3\right )^{6} - 16 \, \log \left (\log \relax (2) + 3\right )^{7} - 2 \, x^{5} e^{x} + 84 \, x \log \left (\log \relax (2) + 3\right )^{5} + 112 \, \log \left (\log \relax (2) + 3\right )^{6} - 280 \, x \log \left (\log \relax (2) + 3\right )^{4} - 448 \, \log \left (\log \relax (2) + 3\right )^{5} + 560 \, x \log \left (\log \relax (2) + 3\right )^{3} + 1120 \, \log \left (\log \relax (2) + 3\right )^{4} - 672 \, x \log \left (\log \relax (2) + 3\right )^{2} - 1792 \, \log \left (\log \relax (2) + 3\right )^{3} + 448 \, x \log \left (\log \relax (2) + 3\right ) + 1792 \, \log \left (\log \relax (2) + 3\right )^{2} - 128 \, x - 1024 \, \log \left (\log \relax (2) + 3\right ) + 256}{x + \log \left (\log \relax (2) + 3\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x)+8*x^7)*log(3+log(2))+(x^7+x^6-16*x^
5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm="giac")

[Out]

(x^8 + x*log(log(2) + 3)^7 + log(log(2) + 3)^8 + x^6*e^x + x^5*e^x*log(log(2) + 3) - 14*x*log(log(2) + 3)^6 -
16*log(log(2) + 3)^7 - 2*x^5*e^x + 84*x*log(log(2) + 3)^5 + 112*log(log(2) + 3)^6 - 280*x*log(log(2) + 3)^4 -
448*log(log(2) + 3)^5 + 560*x*log(log(2) + 3)^3 + 1120*log(log(2) + 3)^4 - 672*x*log(log(2) + 3)^2 - 1792*log(
log(2) + 3)^3 + 448*x*log(log(2) + 3) + 1792*log(log(2) + 3)^2 - 128*x - 1024*log(log(2) + 3) + 256)/(x + log(
log(2) + 3) - 2)

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maple [A]  time = 0.62, size = 35, normalized size = 1.59

method	result	size
norman	\(\frac {x^{8}+x^{6} {\mathrm e}^{x}+\left (\ln \left (3+\ln \relax (2)\right )-2\right ) x^{5} {\mathrm e}^{x}}{x -2+\ln \left (3+\ln \relax (2)\right )}\)	\(35\)
default	\(64 x +x^{5} {\mathrm e}^{x}+x^{7}+2 x^{6}+4 x^{5}+8 x^{4}+16 x^{3}+32 x^{2}+\frac {256}{x -2+\ln \left (3+\ln \relax (2)\right )}+\ln \left (3+\ln \relax (2)\right )^{6} x +\frac {\ln \left (3+\ln \relax (2)\right )^{8}}{x -2+\ln \left (3+\ln \relax (2)\right )}+10 \ln \left (3+\ln \relax (2)\right )^{4} x^{2}-8 \ln \left (3+\ln \relax (2)\right )^{3} x^{3}+6 \ln \left (3+\ln \relax (2)\right )^{2} x^{4}-4 x^{5} \ln \left (3+\ln \relax (2)\right )-\frac {16 \ln \left (3+\ln \relax (2)\right )^{7}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {112 \ln \left (3+\ln \relax (2)\right )^{6}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {448 \ln \left (3+\ln \relax (2)\right )^{5}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {1120 \ln \left (3+\ln \relax (2)\right )^{4}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {1792 \ln \left (3+\ln \relax (2)\right )^{3}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {1792 \ln \left (3+\ln \relax (2)\right )^{2}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {1024 \ln \left (3+\ln \relax (2)\right )}{x -2+\ln \left (3+\ln \relax (2)\right )}-40 \ln \left (3+\ln \relax (2)\right )^{3} x^{2}-12 \ln \left (3+\ln \relax (2)\right )^{5} x +60 \ln \left (3+\ln \relax (2)\right )^{4} x -160 \ln \left (3+\ln \relax (2)\right )^{3} x +240 \ln \left (3+\ln \relax (2)\right )^{2} x +80 \ln \left (3+\ln \relax (2)\right )^{2} x^{2}-80 \ln \left (3+\ln \relax (2)\right ) x^{2}+24 \ln \left (3+\ln \relax (2)\right )^{2} x^{3}-12 \ln \left (3+\ln \relax (2)\right ) x^{4}-32 \ln \left (3+\ln \relax (2)\right ) x^{3}-192 \ln \left (3+\ln \relax (2)\right ) x +\ln \left (3+\ln \relax (2)\right )^{4} x^{3}-\ln \left (3+\ln \relax (2)\right )^{3} x^{4}+\ln \left (3+\ln \relax (2)\right )^{2} x^{5}-x^{6} \ln \left (3+\ln \relax (2)\right )-\ln \left (3+\ln \relax (2)\right )^{5} x^{2}\)	\(425\)
risch	\(64 x +x^{5} {\mathrm e}^{x}+x^{7}+2 x^{6}+4 x^{5}+8 x^{4}+16 x^{3}+32 x^{2}+\frac {256}{x -2+\ln \left (3+\ln \relax (2)\right )}+\ln \left (3+\ln \relax (2)\right )^{6} x +\frac {\ln \left (3+\ln \relax (2)\right )^{8}}{x -2+\ln \left (3+\ln \relax (2)\right )}+10 \ln \left (3+\ln \relax (2)\right )^{4} x^{2}-8 \ln \left (3+\ln \relax (2)\right )^{3} x^{3}+6 \ln \left (3+\ln \relax (2)\right )^{2} x^{4}-4 x^{5} \ln \left (3+\ln \relax (2)\right )-\frac {16 \ln \left (3+\ln \relax (2)\right )^{7}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {112 \ln \left (3+\ln \relax (2)\right )^{6}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {448 \ln \left (3+\ln \relax (2)\right )^{5}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {1120 \ln \left (3+\ln \relax (2)\right )^{4}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {1792 \ln \left (3+\ln \relax (2)\right )^{3}}{x -2+\ln \left (3+\ln \relax (2)\right )}+\frac {1792 \ln \left (3+\ln \relax (2)\right )^{2}}{x -2+\ln \left (3+\ln \relax (2)\right )}-\frac {1024 \ln \left (3+\ln \relax (2)\right )}{x -2+\ln \left (3+\ln \relax (2)\right )}-40 \ln \left (3+\ln \relax (2)\right )^{3} x^{2}-12 \ln \left (3+\ln \relax (2)\right )^{5} x +60 \ln \left (3+\ln \relax (2)\right )^{4} x -160 \ln \left (3+\ln \relax (2)\right )^{3} x +240 \ln \left (3+\ln \relax (2)\right )^{2} x +80 \ln \left (3+\ln \relax (2)\right )^{2} x^{2}-80 \ln \left (3+\ln \relax (2)\right ) x^{2}+24 \ln \left (3+\ln \relax (2)\right )^{2} x^{3}-12 \ln \left (3+\ln \relax (2)\right ) x^{4}-32 \ln \left (3+\ln \relax (2)\right ) x^{3}-192 \ln \left (3+\ln \relax (2)\right ) x +\ln \left (3+\ln \relax (2)\right )^{4} x^{3}-\ln \left (3+\ln \relax (2)\right )^{3} x^{4}+\ln \left (3+\ln \relax (2)\right )^{2} x^{5}-x^{6} \ln \left (3+\ln \relax (2)\right )-\ln \left (3+\ln \relax (2)\right )^{5} x^{2}\)	\(425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5+5*x^4)*exp(x)*ln(3+ln(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x)+8*x^7)*ln(3+ln(2))+(x^7+x^6-16*x^5+20*x^4)*
exp(x)+7*x^8-16*x^7)/(ln(3+ln(2))^2+(2*x-4)*ln(3+ln(2))+x^2-4*x+4),x,method=_RETURNVERBOSE)

[Out]

(x^8+x^6*exp(x)+(ln(3+ln(2))-2)*x^5*exp(x))/(x-2+ln(3+ln(2)))

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maxima [B]  time = 0.80, size = 955, normalized size = 43.41 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5+5*x^4)*exp(x)*log(3+log(2))^2+((2*x^6+6*x^5-20*x^4)*exp(x)+8*x^7)*log(3+log(2))+(x^7+x^6-16*x^
5+20*x^4)*exp(x)+7*x^8-16*x^7)/(log(3+log(2))^2+(2*x-4)*log(3+log(2))+x^2-4*x+4),x, algorithm="maxima")

[Out]

x^7 - 7/3*x^6*(log(log(2) + 3) - 2) + 21/5*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*x^5 - 8/3*x^6 + 32/5*x^
5*(log(log(2) + 3) - 2) + x^5*e^x - 7*(log(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(log(2) + 3) - 8)*x^4 -
12*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*x^4 + 35/3*(log(log(2) + 3)^4 - 8*log(log(2) + 3)^3 + 24*log(l
og(2) + 3)^2 - 32*log(log(2) + 3) + 16)*x^3 + 64/3*(log(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(log(2) +
3) - 8)*x^3 - 21*(log(log(2) + 3)^5 - 10*log(log(2) + 3)^4 + 40*log(log(2) + 3)^3 - 80*log(log(2) + 3)^2 + 80*
log(log(2) + 3) - 32)*x^2 - 40*(log(log(2) + 3)^4 - 8*log(log(2) + 3)^3 + 24*log(log(2) + 3)^2 - 32*log(log(2)
+ 3) + 16)*x^2 + 49*(log(log(2) + 3)^6 - 12*log(log(2) + 3)^5 + 60*log(log(2) + 3)^4 - 160*log(log(2) + 3)^3
+ 240*log(log(2) + 3)^2 - 192*log(log(2) + 3) + 64)*x + 96*(log(log(2) + 3)^5 - 10*log(log(2) + 3)^4 + 40*log(
log(2) + 3)^3 - 80*log(log(2) + 3)^2 + 80*log(log(2) + 3) - 32)*x - 56*(log(log(2) + 3)^7 - 14*log(log(2) + 3)
^6 + 84*log(log(2) + 3)^5 - 280*log(log(2) + 3)^4 + 560*log(log(2) + 3)^3 - 672*log(log(2) + 3)^2 + 448*log(lo
g(2) + 3) - 128)*log(x + log(log(2) + 3) - 2) - 112*(log(log(2) + 3)^6 - 12*log(log(2) + 3)^5 + 60*log(log(2)
+ 3)^4 - 160*log(log(2) + 3)^3 + 240*log(log(2) + 3)^2 - 192*log(log(2) + 3) + 64)*log(x + log(log(2) + 3) - 2
) + 2/15*(10*x^6 - 24*x^5*(log(log(2) + 3) - 2) + 45*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*x^4 - 80*(log
(log(2) + 3)^3 - 6*log(log(2) + 3)^2 + 12*log(log(2) + 3) - 8)*x^3 + 150*(log(log(2) + 3)^4 - 8*log(log(2) + 3
)^3 + 24*log(log(2) + 3)^2 - 32*log(log(2) + 3) + 16)*x^2 - 360*(log(log(2) + 3)^5 - 10*log(log(2) + 3)^4 + 40
*log(log(2) + 3)^3 - 80*log(log(2) + 3)^2 + 80*log(log(2) + 3) - 32)*x + 420*(log(log(2) + 3)^6 - 12*log(log(2
) + 3)^5 + 60*log(log(2) + 3)^4 - 160*log(log(2) + 3)^3 + 240*log(log(2) + 3)^2 - 192*log(log(2) + 3) + 64)*lo
g(x + log(log(2) + 3) - 2) + 60*(log(log(2) + 3)^7 - 14*log(log(2) + 3)^6 + 84*log(log(2) + 3)^5 - 280*log(log
(2) + 3)^4 + 560*log(log(2) + 3)^3 - 672*log(log(2) + 3)^2 + 448*log(log(2) + 3) - 128)/(x + log(log(2) + 3) -
2))*log(log(2) + 3) - 7*(log(log(2) + 3)^8 - 16*log(log(2) + 3)^7 + 112*log(log(2) + 3)^6 - 448*log(log(2) +
3)^5 + 1120*log(log(2) + 3)^4 - 1792*log(log(2) + 3)^3 + 1792*log(log(2) + 3)^2 - 1024*log(log(2) + 3) + 256)/
(x + log(log(2) + 3) - 2) - 16*(log(log(2) + 3)^7 - 14*log(log(2) + 3)^6 + 84*log(log(2) + 3)^5 - 280*log(log(
2) + 3)^4 + 560*log(log(2) + 3)^3 - 672*log(log(2) + 3)^2 + 448*log(log(2) + 3) - 128)/(x + log(log(2) + 3) -
2)

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mupad [B]  time = 4.27, size = 1697, normalized size = 77.14 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2) + 3)*(exp(x)*(6*x^5 - 20*x^4 + 2*x^6) + 8*x^7) + exp(x)*(20*x^4 - 16*x^5 + x^6 + x^7) - 16*x^7
+ 7*x^8 + log(log(2) + 3)^2*exp(x)*(5*x^4 + x^5))/(log(log(2) + 3)^2 - 4*x + log(log(2) + 3)*(2*x - 4) + x^2
+ 4),x)

[Out]

x^5*exp(x) + x*((((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)
^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*lo
g(log(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4) - (((log((log(2) + 3)
^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*
log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^
2) + 12))*(log((log(2) + 3)^2) - 4) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*((log((log(2) + 3)^2) - 4)*(
8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28))*(log((log(2)
+ 3)^2) - 4))*(log((log(2) + 3)^2) - 4) + (((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2
) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4
*log(log(2) + 3) + 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log((log(2) + 3)^2) - 4) - (log(log(2
) + 3)^2 - 4*log(log(2) + 3) + 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12)
- 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28))*(log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)) + x^6*((4*log(l
og(2) + 3))/3 - (7*log((log(2) + 3)^2))/6 + 2) - x^2*((((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(
8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2)
+ 3)^2 - 4*log(log(2) + 3) + 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log(log(2) + 3)^2 - 4*log(
log(2) + 3) + 4))/2 - ((((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(
2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4
)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12))*(log((log(2) + 3)^2) - 4) - (log(log(2) + 3)^2 - 4*log(log
(2) + 3) + 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3)
+ 7*log(log(2) + 3)^2 + 28))*(log((log(2) + 3)^2) - 4))/2) - x^5*(((log((log(2) + 3)^2) - 4)*(8*log(log(2) +
3) - 7*log((log(2) + 3)^2) + 12))/5 - (28*log(log(2) + 3))/5 + (7*log(log(2) + 3)^2)/5 + 28/5) - x^3*((((log((
log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2
) + 3) + 7*log(log(2) + 3)^2 + 28) - (log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*(8*log(log(2) + 3) - 7*log((l
og(2) + 3)^2) + 12))*(log((log(2) + 3)^2) - 4))/3 - ((log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)*((log((log(2)
+ 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28
))/3) + x^7 + x^4*(((log((log(2) + 3)^2) - 4)*((log((log(2) + 3)^2) - 4)*(8*log(log(2) + 3) - 7*log((log(2) +
3)^2) + 12) - 28*log(log(2) + 3) + 7*log(log(2) + 3)^2 + 28))/4 - ((log(log(2) + 3)^2 - 4*log(log(2) + 3) + 4)
*(8*log(log(2) + 3) - 7*log((log(2) + 3)^2) + 12))/4) + (2*atan(((2*x*(log(log(2) + 3) - 2)^6*(12*log(log(2) +
3) - 8*log((log(2) + 3)^2) - 3*log(log(2) + 3)^2 + log((log(2) + 3)^2)^2 + 4))/(8*log((log(2) + 3)^2) - 16*lo
g(log(2) + 3) + 4*log(log(2) + 3)^2 - log((log(2) + 3)^2)^2)^(1/2) + ((log(log(2) + 3) - 2)^6*(log((log(2) + 3
)^2) - 4)*(12*log(log(2) + 3) - 8*log((log(2) + 3)^2) - 3*log(log(2) + 3)^2 + log((log(2) + 3)^2)^2 + 4))/(8*l
og((log(2) + 3)^2) - 16*log(log(2) + 3) + 4*log(log(2) + 3)^2 - log((log(2) + 3)^2)^2)^(1/2))/(512*log((log(2)
+ 3)^2) - 1536*log(log(2) + 3)*log((log(2) + 3)^2) + 192*log(log(2) + 3)*log((log(2) + 3)^2)^2 + 1920*log(log
(2) + 3)^2*log((log(2) + 3)^2) - 1280*log(log(2) + 3)^3*log((log(2) + 3)^2) + 480*log(log(2) + 3)^4*log((log(2
) + 3)^2) - 96*log(log(2) + 3)^5*log((log(2) + 3)^2) + 8*log(log(2) + 3)^6*log((log(2) + 3)^2) + 1536*log(log(
2) + 3)^2 - 2816*log(log(2) + 3)^3 + 2400*log(log(2) + 3)^4 - 1152*log(log(2) + 3)^5 + 320*log(log(2) + 3)^6 -
48*log(log(2) + 3)^7 + 3*log(log(2) + 3)^8 - 240*log(log(2) + 3)^2*log((log(2) + 3)^2)^2 + 160*log(log(2) + 3
)^3*log((log(2) + 3)^2)^2 - 60*log(log(2) + 3)^4*log((log(2) + 3)^2)^2 + 12*log(log(2) + 3)^5*log((log(2) + 3)
^2)^2 - log(log(2) + 3)^6*log((log(2) + 3)^2)^2 - 64*log((log(2) + 3)^2)^2 - 256))*(log(log(2) + 3) - 2)^6*(12
*log(log(2) + 3) - 8*log((log(2) + 3)^2) - 3*log(log(2) + 3)^2 + log((log(2) + 3)^2)^2 + 4))/(8*log((log(2) +
3)^2) - 16*log(log(2) + 3) + 4*log(log(2) + 3)^2 - log((log(2) + 3)^2)^2)^(1/2)

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sympy [B]  time = 0.74, size = 321, normalized size = 14.59 \begin {gather*} x^{7} + x^{6} \left (2 - \log {\left (\log {\relax (2 )} + 3 \right )}\right ) + x^{5} e^{x} + x^{5} \left (- 4 \log {\left (\log {\relax (2 )} + 3 \right )} + \log {\left (\log {\relax (2 )} + 3 \right )}^{2} + 4\right ) + x^{4} \left (- 12 \log {\left (\log {\relax (2 )} + 3 \right )} - \log {\left (\log {\relax (2 )} + 3 \right )}^{3} + 8 + 6 \log {\left (\log {\relax (2 )} + 3 \right )}^{2}\right ) + x^{3} \left (- 32 \log {\left (\log {\relax (2 )} + 3 \right )} - 8 \log {\left (\log {\relax (2 )} + 3 \right )}^{3} + \log {\left (\log {\relax (2 )} + 3 \right )}^{4} + 16 + 24 \log {\left (\log {\relax (2 )} + 3 \right )}^{2}\right ) + x^{2} \left (- 80 \log {\left (\log {\relax (2 )} + 3 \right )} - 40 \log {\left (\log {\relax (2 )} + 3 \right )}^{3} - \log {\left (\log {\relax (2 )} + 3 \right )}^{5} + 10 \log {\left (\log {\relax (2 )} + 3 \right )}^{4} + 32 + 80 \log {\left (\log {\relax (2 )} + 3 \right )}^{2}\right ) + x \left (- 160 \log {\left (\log {\relax (2 )} + 3 \right )}^{3} - 192 \log {\left (\log {\relax (2 )} + 3 \right )} - 12 \log {\left (\log {\relax (2 )} + 3 \right )}^{5} + \log {\left (\log {\relax (2 )} + 3 \right )}^{6} + 64 + 60 \log {\left (\log {\relax (2 )} + 3 \right )}^{4} + 240 \log {\left (\log {\relax (2 )} + 3 \right )}^{2}\right ) + \frac {- 1792 \log {\left (\log {\relax (2 )} + 3 \right )}^{3} - 448 \log {\left (\log {\relax (2 )} + 3 \right )}^{5} - 1024 \log {\left (\log {\relax (2 )} + 3 \right )} - 16 \log {\left (\log {\relax (2 )} + 3 \right )}^{7} + \log {\left (\log {\relax (2 )} + 3 \right )}^{8} + 256 + 112 \log {\left (\log {\relax (2 )} + 3 \right )}^{6} + 1792 \log {\left (\log {\relax (2 )} + 3 \right )}^{2} + 1120 \log {\left (\log {\relax (2 )} + 3 \right )}^{4}}{x - 2 + \log {\left (\log {\relax (2 )} + 3 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5+5*x**4)*exp(x)*ln(3+ln(2))**2+((2*x**6+6*x**5-20*x**4)*exp(x)+8*x**7)*ln(3+ln(2))+(x**7+x**6-
16*x**5+20*x**4)*exp(x)+7*x**8-16*x**7)/(ln(3+ln(2))**2+(2*x-4)*ln(3+ln(2))+x**2-4*x+4),x)

[Out]

x**7 + x**6*(2 - log(log(2) + 3)) + x**5*exp(x) + x**5*(-4*log(log(2) + 3) + log(log(2) + 3)**2 + 4) + x**4*(-
12*log(log(2) + 3) - log(log(2) + 3)**3 + 8 + 6*log(log(2) + 3)**2) + x**3*(-32*log(log(2) + 3) - 8*log(log(2)
+ 3)**3 + log(log(2) + 3)**4 + 16 + 24*log(log(2) + 3)**2) + x**2*(-80*log(log(2) + 3) - 40*log(log(2) + 3)**
3 - log(log(2) + 3)**5 + 10*log(log(2) + 3)**4 + 32 + 80*log(log(2) + 3)**2) + x*(-160*log(log(2) + 3)**3 - 19
2*log(log(2) + 3) - 12*log(log(2) + 3)**5 + log(log(2) + 3)**6 + 64 + 60*log(log(2) + 3)**4 + 240*log(log(2) +
3)**2) + (-1792*log(log(2) + 3)**3 - 448*log(log(2) + 3)**5 - 1024*log(log(2) + 3) - 16*log(log(2) + 3)**7 +
log(log(2) + 3)**8 + 256 + 112*log(log(2) + 3)**6 + 1792*log(log(2) + 3)**2 + 1120*log(log(2) + 3)**4)/(x - 2
+ log(log(2) + 3))

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