\(\int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+(2 x^3-6 x^4+2 x^6) \log (2)+x^4 \log ^2(2)} (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+(6 x^2-24 x^3+12 x^5) \log (2)+4 x^3 \log ^2(2)) \, dx\) [7948]

Optimal result

Integrand size = 114, antiderivative size = 17 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=e^{\left (x+x^2 \left (-3+x^2+\log (2)\right )\right )^2} \]

[Out]

exp((x+x^2*(ln(2)+x^2-3))^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(17)=34\).

Time = 1.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6, 6838} \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=2^{2 x^6-6 x^4+2 x^3} \exp \left (x^8-6 x^6+2 x^5+9 x^4+x^4 \log ^2(2)-6 x^3+x^2\right ) \]

[In]

Int[E^(x^2 - 6*x^3 + 9*x^4 + 2*x^5 - 6*x^6 + x^8 + (2*x^3 - 6*x^4 + 2*x^6)*Log[2] + x^4*Log[2]^2)*(2*x - 18*x^
2 + 36*x^3 + 10*x^4 - 36*x^5 + 8*x^7 + (6*x^2 - 24*x^3 + 12*x^5)*Log[2] + 4*x^3*Log[2]^2),x]

[Out]

2^(2*x^3 - 6*x^4 + 2*x^6)*E^(x^2 - 6*x^3 + 9*x^4 + 2*x^5 - 6*x^6 + x^8 + x^4*Log[2]^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 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int \exp \left (x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)\right ) \left (2 x-18 x^2+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+x^3 \left (36+4 \log ^2(2)\right )\right ) \, dx \\ & = 2^{2 x^3-6 x^4+2 x^6} \exp \left (x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+x^4 \log ^2(2)\right ) \\ \end{align*}

Mathematica [F]

\[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=\int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx \]

[In]

Integrate[E^(x^2 - 6*x^3 + 9*x^4 + 2*x^5 - 6*x^6 + x^8 + (2*x^3 - 6*x^4 + 2*x^6)*Log[2] + x^4*Log[2]^2)*(2*x -
18*x^2 + 36*x^3 + 10*x^4 - 36*x^5 + 8*x^7 + (6*x^2 - 24*x^3 + 12*x^5)*Log[2] + 4*x^3*Log[2]^2),x]

[Out]

Integrate[E^(x^2 - 6*x^3 + 9*x^4 + 2*x^5 - 6*x^6 + x^8 + (2*x^3 - 6*x^4 + 2*x^6)*Log[2] + x^4*Log[2]^2)*(2*x -
18*x^2 + 36*x^3 + 10*x^4 - 36*x^5 + 8*x^7 + (6*x^2 - 24*x^3 + 12*x^5)*Log[2] + 4*x^3*Log[2]^2), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(16)=32\).

Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.06

method	result	size
risch	\(4^{x^{3} \left (x^{3}-3 x +1\right )} {\mathrm e}^{x^{2} \left (x^{6}+x^{2} \ln \left (2\right )^{2}-6 x^{4}+2 x^{3}+9 x^{2}-6 x +1\right )}\)	\(52\)
derivativedivides	\({\mathrm e}^{x^{4} \ln \left (2\right )^{2}+\left (2 x^{6}-6 x^{4}+2 x^{3}\right ) \ln \left (2\right )+x^{8}-6 x^{6}+2 x^{5}+9 x^{4}-6 x^{3}+x^{2}}\)	\(56\)
default	\({\mathrm e}^{x^{4} \ln \left (2\right )^{2}+\left (2 x^{6}-6 x^{4}+2 x^{3}\right ) \ln \left (2\right )+x^{8}-6 x^{6}+2 x^{5}+9 x^{4}-6 x^{3}+x^{2}}\)	\(56\)
norman	\({\mathrm e}^{x^{4} \ln \left (2\right )^{2}+\left (2 x^{6}-6 x^{4}+2 x^{3}\right ) \ln \left (2\right )+x^{8}-6 x^{6}+2 x^{5}+9 x^{4}-6 x^{3}+x^{2}}\)	\(56\)
parallelrisch	\({\mathrm e}^{x^{4} \ln \left (2\right )^{2}+\left (2 x^{6}-6 x^{4}+2 x^{3}\right ) \ln \left (2\right )+x^{8}-6 x^{6}+2 x^{5}+9 x^{4}-6 x^{3}+x^{2}}\)	\(56\)
gosper	\({\mathrm e}^{x^{8}+2 x^{6} \ln \left (2\right )+x^{4} \ln \left (2\right )^{2}-6 x^{6}-6 x^{4} \ln \left (2\right )+2 x^{5}+2 x^{3} \ln \left (2\right )+9 x^{4}-6 x^{3}+x^{2}}\)	\(58\)

[In]

int((4*x^3*ln(2)^2+(12*x^5-24*x^3+6*x^2)*ln(2)+8*x^7-36*x^5+10*x^4+36*x^3-18*x^2+2*x)*exp(x^4*ln(2)^2+(2*x^6-6
*x^4+2*x^3)*ln(2)+x^8-6*x^6+2*x^5+9*x^4-6*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

4^(x^3*(x^3-3*x+1))*exp(x^2*(x^6+x^2*ln(2)^2-6*x^4+2*x^3+9*x^2-6*x+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.06 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=e^{\left (x^{8} - 6 \, x^{6} + x^{4} \log \left (2\right )^{2} + 2 \, x^{5} + 9 \, x^{4} - 6 \, x^{3} + x^{2} + 2 \, {\left (x^{6} - 3 \, x^{4} + x^{3}\right )} \log \left (2\right )\right )} \]

[In]

integrate((4*x^3*log(2)^2+(12*x^5-24*x^3+6*x^2)*log(2)+8*x^7-36*x^5+10*x^4+36*x^3-18*x^2+2*x)*exp(x^4*log(2)^2
+(2*x^6-6*x^4+2*x^3)*log(2)+x^8-6*x^6+2*x^5+9*x^4-6*x^3+x^2),x, algorithm="fricas")

[Out]

e^(x^8 - 6*x^6 + x^4*log(2)^2 + 2*x^5 + 9*x^4 - 6*x^3 + x^2 + 2*(x^6 - 3*x^4 + x^3)*log(2))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).

Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.18 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=e^{x^{8} - 6 x^{6} + 2 x^{5} + x^{4} \log {\left (2 \right )}^{2} + 9 x^{4} - 6 x^{3} + x^{2} + \left (2 x^{6} - 6 x^{4} + 2 x^{3}\right ) \log {\left (2 \right )}} \]

[In]

integrate((4*x**3*ln(2)**2+(12*x**5-24*x**3+6*x**2)*ln(2)+8*x**7-36*x**5+10*x**4+36*x**3-18*x**2+2*x)*exp(x**4
*ln(2)**2+(2*x**6-6*x**4+2*x**3)*ln(2)+x**8-6*x**6+2*x**5+9*x**4-6*x**3+x**2),x)

[Out]

exp(x**8 - 6*x**6 + 2*x**5 + x**4*log(2)**2 + 9*x**4 - 6*x**3 + x**2 + (2*x**6 - 6*x**4 + 2*x**3)*log(2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (16) = 32\).

Time = 0.43 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.35 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=e^{\left (x^{8} + 2 \, x^{6} \log \left (2\right ) - 6 \, x^{6} + x^{4} \log \left (2\right )^{2} + 2 \, x^{5} - 6 \, x^{4} \log \left (2\right ) + 9 \, x^{4} + 2 \, x^{3} \log \left (2\right ) - 6 \, x^{3} + x^{2}\right )} \]

[In]

integrate((4*x^3*log(2)^2+(12*x^5-24*x^3+6*x^2)*log(2)+8*x^7-36*x^5+10*x^4+36*x^3-18*x^2+2*x)*exp(x^4*log(2)^2
+(2*x^6-6*x^4+2*x^3)*log(2)+x^8-6*x^6+2*x^5+9*x^4-6*x^3+x^2),x, algorithm="maxima")

[Out]

e^(x^8 + 2*x^6*log(2) - 6*x^6 + x^4*log(2)^2 + 2*x^5 - 6*x^4*log(2) + 9*x^4 + 2*x^3*log(2) - 6*x^3 + x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.35 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=e^{\left (x^{8} + 2 \, x^{6} \log \left (2\right ) - 6 \, x^{6} + x^{4} \log \left (2\right )^{2} + 2 \, x^{5} - 6 \, x^{4} \log \left (2\right ) + 9 \, x^{4} + 2 \, x^{3} \log \left (2\right ) - 6 \, x^{3} + x^{2}\right )} \]

[In]

integrate((4*x^3*log(2)^2+(12*x^5-24*x^3+6*x^2)*log(2)+8*x^7-36*x^5+10*x^4+36*x^3-18*x^2+2*x)*exp(x^4*log(2)^2
+(2*x^6-6*x^4+2*x^3)*log(2)+x^8-6*x^6+2*x^5+9*x^4-6*x^3+x^2),x, algorithm="giac")

[Out]

e^(x^8 + 2*x^6*log(2) - 6*x^6 + x^4*log(2)^2 + 2*x^5 - 6*x^4*log(2) + 9*x^4 + 2*x^3*log(2) - 6*x^3 + x^2)

Mupad [B] (verification not implemented)

Time = 13.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.82 \[ \int e^{x^2-6 x^3+9 x^4+2 x^5-6 x^6+x^8+\left (2 x^3-6 x^4+2 x^6\right ) \log (2)+x^4 \log ^2(2)} \left (2 x-18 x^2+36 x^3+10 x^4-36 x^5+8 x^7+\left (6 x^2-24 x^3+12 x^5\right ) \log (2)+4 x^3 \log ^2(2)\right ) \, dx=\frac {2^{2\,x^3}\,2^{2\,x^6}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^8}\,{\mathrm {e}}^{x^4\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{2\,x^5}\,{\mathrm {e}}^{-6\,x^3}\,{\mathrm {e}}^{-6\,x^6}\,{\mathrm {e}}^{9\,x^4}}{2^{6\,x^4}} \]

[In]

int(exp(x^4*log(2)^2 + log(2)*(2*x^3 - 6*x^4 + 2*x^6) + x^2 - 6*x^3 + 9*x^4 + 2*x^5 - 6*x^6 + x^8)*(2*x + 4*x^
3*log(2)^2 + log(2)*(6*x^2 - 24*x^3 + 12*x^5) - 18*x^2 + 36*x^3 + 10*x^4 - 36*x^5 + 8*x^7),x)

[Out]

(2^(2*x^3)*2^(2*x^6)*exp(x^2)*exp(x^8)*exp(x^4*log(2)^2)*exp(2*x^5)*exp(-6*x^3)*exp(-6*x^6)*exp(9*x^4))/2^(6*x
^4)