3.75.99 \(\int e^{1+4 x+6 x^2+4 x^3+x^4+(-4-12 x-12 x^2-4 x^3) \log ((8+i \pi )^2)+(6+12 x+6 x^2) \log ^2((8+i \pi )^2)+(-4-4 x) \log ^3((8+i \pi )^2)+\log ^4((8+i \pi )^2)} (20+60 x+60 x^2+20 x^3+(-60-120 x-60 x^2) \log ((8+i \pi )^2)+(60+60 x) \log ^2((8+i \pi )^2)-20 \log ^3((8+i \pi )^2)) \, dx\)

Optimal. Leaf size=21 \[ 5 e^{\left (-1-x+\log \left ((8+i \pi )^2\right )\right )^4} \]

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Rubi [A]  time = 0.34, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 168, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2227, 2226, 12, 2209} \begin {gather*} 5 e^{\left (x+1-\log \left ((8+i \pi )^2\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)*Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^2)*Log
[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Log[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 + (-60 -
120*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Log[(8 + I*Pi)^2]^3),x]

[Out]

5*E^(1 + x - Log[(8 + I*Pi)^2])^4

Rule 12

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

Rule 2209

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

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{\left (1+x-\log \left ((8+i \pi )^2\right )\right )^4} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx\\ &=\int 20 e^{\left (1+x-\log \left ((8+i \pi )^2\right )\right )^4} \left (1+x-\log \left ((8+i \pi )^2\right )\right )^3 \, dx\\ &=20 \int e^{\left (1+x-\log \left ((8+i \pi )^2\right )\right )^4} \left (1+x-\log \left ((8+i \pi )^2\right )\right )^3 \, dx\\ &=5 e^{\left (1+x-\log \left ((8+i \pi )^2\right )\right )^4}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 4.50, size = 0, normalized size = 0.00 \begin {gather*} \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)*Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^
2)*Log[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Log[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 +
(-60 - 120*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Log[(8 + I*Pi)^2]^3),x]

[Out]

Integrate[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)*Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^
2)*Log[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Log[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 +
(-60 - 120*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Log[(8 + I*Pi)^2]^3), x]

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fricas [C]  time = 1.00, size = 255, normalized size = 12.14 \begin {gather*} 5 \, \cosh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x + 8 \, \log \left (i \, \pi + 8\right ) - 1\right ) - 5 \, \sinh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x + 8 \, \log \left (i \, \pi + 8\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2*(-60*x^2-120*x-60)*log(log(-exp(4)^2
))+20*x^3+60*x^2+60*x+20)*exp(16*log(log(-exp(4)^2))^4+8*(-4*x-4)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(l
og(-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1),x, algorithm="fricas")

[Out]

5*cosh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3
+ 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi
+ 8)^2 - 4*x + 8*log(I*pi + 8) - 1) - 5*sinh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I
*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6
*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi + 8)^2 - 4*x + 8*log(I*pi + 8) - 1)

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giac [B]  time = 0.23, size = 125, normalized size = 5.95 \begin {gather*} 5 \, e^{\left (x^{4} - 8 \, x^{3} \log \left (\log \left (-e^{8}\right )\right ) + 24 \, x^{2} \log \left (\log \left (-e^{8}\right )\right )^{2} - 32 \, x \log \left (\log \left (-e^{8}\right )\right )^{3} + 16 \, \log \left (\log \left (-e^{8}\right )\right )^{4} + 4 \, x^{3} - 24 \, x^{2} \log \left (\log \left (-e^{8}\right )\right ) + 48 \, x \log \left (\log \left (-e^{8}\right )\right )^{2} - 32 \, \log \left (\log \left (-e^{8}\right )\right )^{3} + 6 \, x^{2} - 24 \, x \log \left (\log \left (-e^{8}\right )\right ) + 24 \, \log \left (\log \left (-e^{8}\right )\right )^{2} + 4 \, x - 8 \, \log \left (\log \left (-e^{8}\right )\right ) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2*(-60*x^2-120*x-60)*log(log(-exp(4)^2
))+20*x^3+60*x^2+60*x+20)*exp(16*log(log(-exp(4)^2))^4+8*(-4*x-4)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(l
og(-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1),x, algorithm="giac")

[Out]

5*e^(x^4 - 8*x^3*log(log(-e^8)) + 24*x^2*log(log(-e^8))^2 - 32*x*log(log(-e^8))^3 + 16*log(log(-e^8))^4 + 4*x^
3 - 24*x^2*log(log(-e^8)) + 48*x*log(log(-e^8))^2 - 32*log(log(-e^8))^3 + 6*x^2 - 24*x*log(log(-e^8)) + 24*log
(log(-e^8))^2 + 4*x - 8*log(log(-e^8)) + 1)

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maple [B]  time = 10.51, size = 98, normalized size = 4.67

method	result	size
norman	\(5 \,{\mathrm e}^{16 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{4}+8 \left (-4 x -4\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{3}+4 \left (6 x^{2}+12 x +6\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{2}+2 \left (-4 x^{3}-12 x^{2}-12 x -4\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+x^{4}+4 x^{3}+6 x^{2}+4 x +1}\)	\(98\)
risch	\(5 \left (i \pi +8\right )^{-8 \left (x +1\right )^{3}} {\mathrm e}^{16 \ln \left (i \pi +8\right )^{4}-32 \ln \left (i \pi +8\right )^{3} x +24 \ln \left (i \pi +8\right )^{2} x^{2}+x^{4}-32 \ln \left (i \pi +8\right )^{3}+48 \ln \left (i \pi +8\right )^{2} x +4 x^{3}+24 \ln \left (i \pi +8\right )^{2}+6 x^{2}+4 x +1}\)	\(107\)
gosper	\(5 \,{\mathrm e}^{16 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{4}-32 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{3} x +24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{2} x^{2}-8 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x^{3}+x^{4}-32 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{3}+48 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{2} x -24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x^{2}+4 x^{3}+24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )^{2}-24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x +6 x^{2}-8 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+4 x +1}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-160*ln(ln(-exp(4)^2))^3+4*(60*x+60)*ln(ln(-exp(4)^2))^2+2*(-60*x^2-120*x-60)*ln(ln(-exp(4)^2))+20*x^3+60
*x^2+60*x+20)*exp(16*ln(ln(-exp(4)^2))^4+8*(-4*x-4)*ln(ln(-exp(4)^2))^3+4*(6*x^2+12*x+6)*ln(ln(-exp(4)^2))^2+2
*(-4*x^3-12*x^2-12*x-4)*ln(ln(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1),x,method=_RETURNVERBOSE)

[Out]

5*exp(16*ln(ln(-exp(4)^2))^4+8*(-4*x-4)*ln(ln(-exp(4)^2))^3+4*(6*x^2+12*x+6)*ln(ln(-exp(4)^2))^2+2*(-4*x^3-12*
x^2-12*x-4)*ln(ln(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1)

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maxima [C]  time = 0.74, size = 281, normalized size = 13.38 \begin {gather*} \frac {20 \, {\left (\cosh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x - 1\right ) - \sinh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x - 1\right )\right )}}{67108864 i \, \pi + 4 \, \pi ^{8} - 256 i \, \pi ^{7} - 7168 \, \pi ^{6} + 114688 i \, \pi ^{5} + 1146880 \, \pi ^{4} - 7340032 i \, \pi ^{3} - 29360128 \, \pi ^{2} + 67108864} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2*(-60*x^2-120*x-60)*log(log(-exp(4)^2
))+20*x^3+60*x^2+60*x+20)*exp(16*log(log(-exp(4)^2))^4+8*(-4*x-4)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(l
og(-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1),x, algorithm="maxima")

[Out]

20*(cosh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x
^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*
pi + 8)^2 - 4*x - 1) - sinh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I*pi + 8)^3 - 16*lo
g(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*
pi + 8) - 24*log(I*pi + 8)^2 - 4*x - 1))/(67108864*I*pi + 4*pi^8 - 256*I*pi^7 - 7168*pi^6 + 114688*I*pi^5 + 11
46880*pi^4 - 7340032*I*pi^3 - 29360128*pi^2 + 67108864)

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mupad [B]  time = 5.03, size = 168, normalized size = 8.00 \begin {gather*} \frac {5\,{\mathrm {e}}^{-32\,x\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^3}\,{\mathrm {e}}^{48\,x\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^4}\,\mathrm {e}\,{\mathrm {e}}^{24\,x^2\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{4\,x^3}\,{\mathrm {e}}^{6\,x^2}\,{\mathrm {e}}^{16\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^4}\,{\mathrm {e}}^{24\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{-32\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^3}}{{\left (8+\pi \,1{}\mathrm {i}\right )}^{8\,x^3+24\,x^2+24\,x}\,\left (286720\,\pi ^4-7340032\,\pi ^2-1792\,\pi ^6+\pi ^8+16777216+\pi \,16777216{}\mathrm {i}-\pi ^3\,1835008{}\mathrm {i}+\pi ^5\,28672{}\mathrm {i}-\pi ^7\,64{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x - 8*log(log(-exp(8)))^3*(4*x + 4) - 2*log(log(-exp(8)))*(12*x + 12*x^2 + 4*x^3 + 4) + 4*log(log(-e
xp(8)))^2*(12*x + 6*x^2 + 6) + 6*x^2 + 4*x^3 + x^4 + 16*log(log(-exp(8)))^4 + 1)*(60*x - 2*log(log(-exp(8)))*(
120*x + 60*x^2 + 60) + 4*log(log(-exp(8)))^2*(60*x + 60) + 60*x^2 + 20*x^3 - 160*log(log(-exp(8)))^3 + 20),x)

[Out]

(5*exp(-32*x*log(pi*1i + 8)^3)*exp(48*x*log(pi*1i + 8)^2)*exp(4*x)*exp(x^4)*exp(1)*exp(24*x^2*log(pi*1i + 8)^2
)*exp(4*x^3)*exp(6*x^2)*exp(16*log(pi*1i + 8)^4)*exp(24*log(pi*1i + 8)^2)*exp(-32*log(pi*1i + 8)^3))/((pi*1i +
8)^(24*x + 24*x^2 + 8*x^3)*(pi*16777216i - 7340032*pi^2 - pi^3*1835008i + 286720*pi^4 + pi^5*28672i - 1792*pi
^6 - pi^7*64i + pi^8 + 16777216))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-160*ln(ln(-exp(4)**2))**3+4*(60*x+60)*ln(ln(-exp(4)**2))**2+2*(-60*x**2-120*x-60)*ln(ln(-exp(4)**2
))+20*x**3+60*x**2+60*x+20)*exp(16*ln(ln(-exp(4)**2))**4+8*(-4*x-4)*ln(ln(-exp(4)**2))**3+4*(6*x**2+12*x+6)*ln
(ln(-exp(4)**2))**2+2*(-4*x**3-12*x**2-12*x-4)*ln(ln(-exp(4)**2))+x**4+4*x**3+6*x**2+4*x+1),x)

[Out]

Timed out

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