3.42.94 \(\int \frac {e^{x^2} (e^8 (2 x^4+2 x^6)+(e^{12} (4 x^3+8 x^5)+e^8 (8 x^4+8 x^6)) \log (4)+(12 e^{16} x^4+e^{12} (12 x^3+24 x^5)+e^8 (12 x^4+12 x^6)) \log ^2(4)+(24 e^{16} x^4+e^{20} (-4 x+8 x^3)+e^{12} (12 x^3+24 x^5)+e^8 (8 x^4+8 x^6)) \log ^3(4)+(12 e^{16} x^4+e^{24} (-2+2 x^2)+e^{20} (-4 x+8 x^3)+e^{12} (4 x^3+8 x^5)+e^8 (2 x^4+2 x^6)) \log ^4(4))}{x^3 \log ^4(4)} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^{8+x^2} \left (e^4+x+\frac {x}{\log (4)}\right )^4}{x^2} \]

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Rubi [A]  time = 0.99, antiderivative size = 48, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 3, integrand size = 232, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {12, 6688, 2288} \begin {gather*} \frac {e^{x^2+8} \left (x (1+\log (4))+e^4 \log (4)\right )^3 \left (x^3 (1+\log (4))+e^4 x^2 \log (4)\right )}{x^4 \log ^4(4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x^2*(E^8*(2*x^4 + 2*x^6) + (E^12*(4*x^3 + 8*x^5) + E^8*(8*x^4 + 8*x^6))*Log[4] + (12*E^16*x^4 + E^12*(1
2*x^3 + 24*x^5) + E^8*(12*x^4 + 12*x^6))*Log[4]^2 + (24*E^16*x^4 + E^20*(-4*x + 8*x^3) + E^12*(12*x^3 + 24*x^5
) + E^8*(8*x^4 + 8*x^6))*Log[4]^3 + (12*E^16*x^4 + E^24*(-2 + 2*x^2) + E^20*(-4*x + 8*x^3) + E^12*(4*x^3 + 8*x
^5) + E^8*(2*x^4 + 2*x^6))*Log[4]^4))/(x^3*Log[4]^4),x]

[Out]

(E^(8 + x^2)*(E^4*Log[4] + x*(1 + Log[4]))^3*(E^4*x^2*Log[4] + x^3*(1 + Log[4])))/(x^4*Log[4]^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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

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} &=\frac {\int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3} \, dx}{\log ^4(4)}\\ &=\frac {\int \frac {2 e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (-e^4 \log (4)+e^4 x^2 \log (4)+x (1+\log (4))+x^3 (1+\log (4))\right )}{x^3} \, dx}{\log ^4(4)}\\ &=\frac {2 \int \frac {e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (-e^4 \log (4)+e^4 x^2 \log (4)+x (1+\log (4))+x^3 (1+\log (4))\right )}{x^3} \, dx}{\log ^4(4)}\\ &=\frac {e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (e^4 x^2 \log (4)+x^3 (1+\log (4))\right )}{x^4 \log ^4(4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 29, normalized size = 1.21 \begin {gather*} \frac {e^{8+x^2} \left (x+e^4 \log (4)+x \log (4)\right )^4}{x^2 \log ^4(4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^2*(E^8*(2*x^4 + 2*x^6) + (E^12*(4*x^3 + 8*x^5) + E^8*(8*x^4 + 8*x^6))*Log[4] + (12*E^16*x^4 + E
^12*(12*x^3 + 24*x^5) + E^8*(12*x^4 + 12*x^6))*Log[4]^2 + (24*E^16*x^4 + E^20*(-4*x + 8*x^3) + E^12*(12*x^3 +
24*x^5) + E^8*(8*x^4 + 8*x^6))*Log[4]^3 + (12*E^16*x^4 + E^24*(-2 + 2*x^2) + E^20*(-4*x + 8*x^3) + E^12*(4*x^3
+ 8*x^5) + E^8*(2*x^4 + 2*x^6))*Log[4]^4))/(x^3*Log[4]^4),x]

[Out]

(E^(8 + x^2)*(x + E^4*Log[4] + x*Log[4])^4)/(x^2*Log[4]^4)

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fricas [B]  time = 1.48, size = 128, normalized size = 5.33 \begin {gather*} \frac {{\left (x^{4} e^{8} + 16 \, {\left (x^{4} e^{8} + 4 \, x^{3} e^{12} + 6 \, x^{2} e^{16} + 4 \, x e^{20} + e^{24}\right )} \log \relax (2)^{4} + 32 \, {\left (x^{4} e^{8} + 3 \, x^{3} e^{12} + 3 \, x^{2} e^{16} + x e^{20}\right )} \log \relax (2)^{3} + 24 \, {\left (x^{4} e^{8} + 2 \, x^{3} e^{12} + x^{2} e^{16}\right )} \log \relax (2)^{2} + 8 \, {\left (x^{4} e^{8} + x^{3} e^{12}\right )} \log \relax (2)\right )} e^{\left (x^{2}\right )}}{16 \, x^{2} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+
4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4
)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^
3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)
*log(2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="fricas")

[Out]

1/16*(x^4*e^8 + 16*(x^4*e^8 + 4*x^3*e^12 + 6*x^2*e^16 + 4*x*e^20 + e^24)*log(2)^4 + 32*(x^4*e^8 + 3*x^3*e^12 +
3*x^2*e^16 + x*e^20)*log(2)^3 + 24*(x^4*e^8 + 2*x^3*e^12 + x^2*e^16)*log(2)^2 + 8*(x^4*e^8 + x^3*e^12)*log(2)
)*e^(x^2)/(x^2*log(2)^4)

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giac [B]  time = 0.32, size = 219, normalized size = 9.12 \begin {gather*} \frac {16 \, x^{4} e^{\left (x^{2} + 8\right )} \log \relax (2)^{4} + 32 \, x^{4} e^{\left (x^{2} + 8\right )} \log \relax (2)^{3} + 64 \, x^{3} e^{\left (x^{2} + 12\right )} \log \relax (2)^{4} + 24 \, x^{4} e^{\left (x^{2} + 8\right )} \log \relax (2)^{2} + 96 \, x^{3} e^{\left (x^{2} + 12\right )} \log \relax (2)^{3} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \relax (2)^{4} + 8 \, x^{4} e^{\left (x^{2} + 8\right )} \log \relax (2) + 48 \, x^{3} e^{\left (x^{2} + 12\right )} \log \relax (2)^{2} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \relax (2)^{3} + 64 \, x e^{\left (x^{2} + 20\right )} \log \relax (2)^{4} + x^{4} e^{\left (x^{2} + 8\right )} + 8 \, x^{3} e^{\left (x^{2} + 12\right )} \log \relax (2) + 24 \, x^{2} e^{\left (x^{2} + 16\right )} \log \relax (2)^{2} + 32 \, x e^{\left (x^{2} + 20\right )} \log \relax (2)^{3} + 16 \, e^{\left (x^{2} + 24\right )} \log \relax (2)^{4}}{16 \, x^{2} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+
4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4
)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^
3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)
*log(2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="giac")

[Out]

1/16*(16*x^4*e^(x^2 + 8)*log(2)^4 + 32*x^4*e^(x^2 + 8)*log(2)^3 + 64*x^3*e^(x^2 + 12)*log(2)^4 + 24*x^4*e^(x^2
+ 8)*log(2)^2 + 96*x^3*e^(x^2 + 12)*log(2)^3 + 96*x^2*e^(x^2 + 16)*log(2)^4 + 8*x^4*e^(x^2 + 8)*log(2) + 48*x
^3*e^(x^2 + 12)*log(2)^2 + 96*x^2*e^(x^2 + 16)*log(2)^3 + 64*x*e^(x^2 + 20)*log(2)^4 + x^4*e^(x^2 + 8) + 8*x^3
*e^(x^2 + 12)*log(2) + 24*x^2*e^(x^2 + 16)*log(2)^2 + 32*x*e^(x^2 + 20)*log(2)^3 + 16*e^(x^2 + 24)*log(2)^4)/(
x^2*log(2)^4)

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maple [B]  time = 0.12, size = 155, normalized size = 6.46

method	result	size
risch	\(\frac {\left (16 \ln \relax (2)^{4} {\mathrm e}^{16}+64 \,{\mathrm e}^{12} \ln \relax (2)^{4} x +32 \,{\mathrm e}^{12} \ln \relax (2)^{3} x +96 \,{\mathrm e}^{8} \ln \relax (2)^{4} x^{2}+96 \,{\mathrm e}^{8} \ln \relax (2)^{3} x^{2}+64 \,{\mathrm e}^{4} \ln \relax (2)^{4} x^{3}+24 \,{\mathrm e}^{8} \ln \relax (2)^{2} x^{2}+96 \,{\mathrm e}^{4} \ln \relax (2)^{3} x^{3}+16 \ln \relax (2)^{4} x^{4}+48 \,{\mathrm e}^{4} \ln \relax (2)^{2} x^{3}+32 x^{4} \ln \relax (2)^{3}+8 x^{3} {\mathrm e}^{4} \ln \relax (2)+24 x^{4} \ln \relax (2)^{2}+8 x^{4} \ln \relax (2)+x^{4}\right ) {\mathrm e}^{x^{2}+8}}{16 x^{2} \ln \relax (2)^{4}}\)	\(155\)
norman	\(\frac {{\mathrm e}^{8} {\mathrm e}^{16} \ln \relax (2)^{3} {\mathrm e}^{x^{2}}+\frac {{\mathrm e}^{8} {\mathrm e}^{4} \left (8 \ln \relax (2)^{3}+12 \ln \relax (2)^{2}+6 \ln \relax (2)+1\right ) x^{3} {\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{8} \left (16 \ln \relax (2)^{4}+32 \ln \relax (2)^{3}+24 \ln \relax (2)^{2}+8 \ln \relax (2)+1\right ) x^{4} {\mathrm e}^{x^{2}}}{16 \ln \relax (2)}+2 \,{\mathrm e}^{8} {\mathrm e}^{12} \ln \relax (2)^{2} \left (1+2 \ln \relax (2)\right ) x \,{\mathrm e}^{x^{2}}+\frac {3 \ln \relax (2) \left ({\mathrm e}^{8}\right )^{2} \left (4 \ln \relax (2)^{2}+4 \ln \relax (2)+1\right ) x^{2} {\mathrm e}^{x^{2}}}{2}}{x^{2} \ln \relax (2)^{3}}\)	\(157\)
gosper	\(\frac {{\mathrm e}^{8} \left (16 \ln \relax (2)^{4} {\mathrm e}^{16}+64 \,{\mathrm e}^{12} \ln \relax (2)^{4} x +32 \,{\mathrm e}^{12} \ln \relax (2)^{3} x +96 \,{\mathrm e}^{8} \ln \relax (2)^{4} x^{2}+96 \,{\mathrm e}^{8} \ln \relax (2)^{3} x^{2}+64 \,{\mathrm e}^{4} \ln \relax (2)^{4} x^{3}+24 \,{\mathrm e}^{8} \ln \relax (2)^{2} x^{2}+96 \,{\mathrm e}^{4} \ln \relax (2)^{3} x^{3}+16 \ln \relax (2)^{4} x^{4}+48 \,{\mathrm e}^{4} \ln \relax (2)^{2} x^{3}+32 x^{4} \ln \relax (2)^{3}+8 x^{3} {\mathrm e}^{4} \ln \relax (2)+24 x^{4} \ln \relax (2)^{2}+8 x^{4} \ln \relax (2)+x^{4}\right ) {\mathrm e}^{x^{2}}}{16 \ln \relax (2)^{4} x^{2}}\)	\(169\)
default	\(\frac {{\mathrm e}^{8} {\mathrm e}^{x^{2}}+2 \,{\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+8 \ln \relax (2) {\mathrm e}^{8} {\mathrm e}^{x^{2}} x^{2}+24 \ln \relax (2)^{2} {\mathrm e}^{8} {\mathrm e}^{x^{2}} x^{2}+32 \ln \relax (2)^{3} {\mathrm e}^{8} {\mathrm e}^{x^{2}} x^{2}+16 \ln \relax (2)^{4} {\mathrm e}^{8} {\mathrm e}^{x^{2}} x^{2}+\frac {16 \ln \relax (2)^{4} {\mathrm e}^{8} {\mathrm e}^{16} {\mathrm e}^{x^{2}}}{x^{2}}+\frac {32 \ln \relax (2)^{3} {\mathrm e}^{8} {\mathrm e}^{12} {\mathrm e}^{x^{2}}}{x}+\frac {64 \ln \relax (2)^{4} {\mathrm e}^{8} {\mathrm e}^{12} {\mathrm e}^{x^{2}}}{x}+24 \ln \relax (2)^{2} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+96 \ln \relax (2)^{3} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+96 \ln \relax (2)^{4} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+8 \ln \relax (2) {\mathrm e}^{8} {\mathrm e}^{4} {\mathrm e}^{x^{2}} x +48 \ln \relax (2)^{2} {\mathrm e}^{8} {\mathrm e}^{4} {\mathrm e}^{x^{2}} x +96 \ln \relax (2)^{3} {\mathrm e}^{8} {\mathrm e}^{4} {\mathrm e}^{x^{2}} x +64 \ln \relax (2)^{4} {\mathrm e}^{8} {\mathrm e}^{4} {\mathrm e}^{x^{2}} x}{16 \ln \relax (2)^{4}}\)	\(288\)
meijerg	\({\mathrm e}^{24} \left (\frac {1}{x^{2}}+1-2 \ln \relax (x )-i \pi -\frac {2 x^{2}+2}{2 x^{2}}+\frac {{\mathrm e}^{x^{2}}}{x^{2}}+\ln \left (-x^{2}\right )+\expIntegralEi \left (1, -x^{2}\right )\right )+\frac {\left (128 \ln \relax (2)^{4} {\mathrm e}^{20}+64 \ln \relax (2)^{3} {\mathrm e}^{20}+64 \ln \relax (2)^{4} {\mathrm e}^{12}+96 \ln \relax (2)^{3} {\mathrm e}^{12}+48 \ln \relax (2)^{2} {\mathrm e}^{12}+8 \,{\mathrm e}^{12} \ln \relax (2)\right ) \sqrt {\pi }\, \erfi \relax (x )}{32 \ln \relax (2)^{4}}+\frac {i \left (-64 \ln \relax (2)^{4} {\mathrm e}^{20}-32 \ln \relax (2)^{3} {\mathrm e}^{20}\right ) \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \sqrt {\pi }\, \erfi \relax (x )\right )}{32 \ln \relax (2)^{4}}+\frac {\left (32 \ln \relax (2)^{4} {\mathrm e}^{8}+64 \ln \relax (2)^{3} {\mathrm e}^{8}+48 \ln \relax (2)^{2} {\mathrm e}^{8}+16 \ln \relax (2) {\mathrm e}^{8}+2 \,{\mathrm e}^{8}\right ) \left (1-\frac {\left (-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{2}\right )}{32 \ln \relax (2)^{4}}-\frac {\left (192 \ln \relax (2)^{4} {\mathrm e}^{16}+192 \ln \relax (2)^{3} {\mathrm e}^{16}+48 \,{\mathrm e}^{16} \ln \relax (2)^{2}+32 \ln \relax (2)^{4} {\mathrm e}^{8}+64 \ln \relax (2)^{3} {\mathrm e}^{8}+48 \ln \relax (2)^{2} {\mathrm e}^{8}+16 \ln \relax (2) {\mathrm e}^{8}+2 \,{\mathrm e}^{8}\right ) \left (1-{\mathrm e}^{x^{2}}\right )}{32 \ln \relax (2)^{4}}+{\mathrm e}^{24} \left (2 \ln \relax (x )+i \pi -\ln \left (-x^{2}\right )-\expIntegralEi \left (1, -x^{2}\right )\right )+\frac {i \left (128 \ln \relax (2)^{4} {\mathrm e}^{12}+192 \ln \relax (2)^{3} {\mathrm e}^{12}+96 \ln \relax (2)^{2} {\mathrm e}^{12}+16 \,{\mathrm e}^{12} \ln \relax (2)\right ) \left (-i x \,{\mathrm e}^{x^{2}}+\frac {i \sqrt {\pi }\, \erfi \relax (x )}{2}\right )}{32 \ln \relax (2)^{4}}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)
*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^4)*ln(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^2+(24
*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*ln(2)^3+4*(12*x^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^3)*exp(2
)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*ln(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*ln(2)+(2
*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/ln(2)^4,x,method=_RETURNVERBOSE)

[Out]

1/16/x^2*(16*ln(2)^4*exp(16)+64*exp(12)*ln(2)^4*x+32*exp(12)*ln(2)^3*x+96*exp(8)*ln(2)^4*x^2+96*exp(8)*ln(2)^3
*x^2+64*exp(4)*ln(2)^4*x^3+24*exp(8)*ln(2)^2*x^2+96*exp(4)*ln(2)^3*x^3+16*ln(2)^4*x^4+48*exp(4)*ln(2)^2*x^3+32
*x^4*ln(2)^3+8*x^3*exp(4)*ln(2)+24*x^4*ln(2)^2+8*x^4*ln(2)+x^4)/ln(2)^4*exp(x^2+8)

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maxima [C]  time = 0.43, size = 467, normalized size = 19.46 \begin {gather*} -\frac {64 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{20} \log \relax (2)^{4} + 32 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \relax (2)^{4} - 16 \, {\rm Ei}\left (x^{2}\right ) e^{24} \log \relax (2)^{4} - 16 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \relax (2)^{4} + 16 \, e^{24} \Gamma \left (-1, -x^{2}\right ) \log \relax (2)^{4} + 32 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{20} \log \relax (2)^{3} + 48 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \relax (2)^{3} - \frac {32 \, \sqrt {-x^{2}} e^{20} \Gamma \left (-\frac {1}{2}, -x^{2}\right ) \log \relax (2)^{4}}{x} - 32 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \relax (2)^{3} + 32 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \relax (2)^{4} - 96 \, e^{\left (x^{2} + 16\right )} \log \relax (2)^{4} - 16 \, e^{\left (x^{2} + 8\right )} \log \relax (2)^{4} + 24 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \relax (2)^{2} - \frac {16 \, \sqrt {-x^{2}} e^{20} \Gamma \left (-\frac {1}{2}, -x^{2}\right ) \log \relax (2)^{3}}{x} - 24 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \relax (2)^{2} + 48 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \relax (2)^{3} - 96 \, e^{\left (x^{2} + 16\right )} \log \relax (2)^{3} - 32 \, e^{\left (x^{2} + 8\right )} \log \relax (2)^{3} + 4 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \relax (2) - 8 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \relax (2) + 24 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \relax (2)^{2} - 24 \, e^{\left (x^{2} + 16\right )} \log \relax (2)^{2} - 24 \, e^{\left (x^{2} + 8\right )} \log \relax (2)^{2} - {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} + 4 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \relax (2) - 8 \, e^{\left (x^{2} + 8\right )} \log \relax (2) - e^{\left (x^{2} + 8\right )}}{16 \, \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+
4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4
)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^
3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)
*log(2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="maxima")

[Out]

-1/16*(64*I*sqrt(pi)*erf(I*x)*e^20*log(2)^4 + 32*I*sqrt(pi)*erf(I*x)*e^12*log(2)^4 - 16*Ei(x^2)*e^24*log(2)^4
- 16*(x^2*e^8 - e^8)*e^(x^2)*log(2)^4 + 16*e^24*gamma(-1, -x^2)*log(2)^4 + 32*I*sqrt(pi)*erf(I*x)*e^20*log(2)^
3 + 48*I*sqrt(pi)*erf(I*x)*e^12*log(2)^3 - 32*sqrt(-x^2)*e^20*gamma(-1/2, -x^2)*log(2)^4/x - 32*(x^2*e^8 - e^8
)*e^(x^2)*log(2)^3 + 32*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2)^4 - 96*e^(x^2 + 16)*log(2)^4 - 1
6*e^(x^2 + 8)*log(2)^4 + 24*I*sqrt(pi)*erf(I*x)*e^12*log(2)^2 - 16*sqrt(-x^2)*e^20*gamma(-1/2, -x^2)*log(2)^3/
x - 24*(x^2*e^8 - e^8)*e^(x^2)*log(2)^2 + 48*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2)^3 - 96*e^(x
^2 + 16)*log(2)^3 - 32*e^(x^2 + 8)*log(2)^3 + 4*I*sqrt(pi)*erf(I*x)*e^12*log(2) - 8*(x^2*e^8 - e^8)*e^(x^2)*lo
g(2) + 24*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2)^2 - 24*e^(x^2 + 16)*log(2)^2 - 24*e^(x^2 + 8)*
log(2)^2 - (x^2*e^8 - e^8)*e^(x^2) + 4*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2) - 8*e^(x^2 + 8)*l
og(2) - e^(x^2 + 8))/log(2)^4

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mupad [B]  time = 3.12, size = 159, normalized size = 6.62 \begin {gather*} \frac {{\mathrm {e}}^{x^2}\,\left (3\,{\mathrm {e}}^{16}+12\,{\mathrm {e}}^{16}\,\ln \relax (2)+12\,{\mathrm {e}}^{16}\,{\ln \relax (2)}^2\right )}{2\,{\ln \relax (2)}^2}+\frac {{\mathrm {e}}^{x^2+24}\,{\ln \relax (2)}^4+2\,x\,{\mathrm {e}}^{x^2+20}\,{\ln \relax (2)}^3\,\left (2\,\ln \relax (2)+1\right )}{x^2\,{\ln \relax (2)}^4}+\frac {x\,{\mathrm {e}}^{x^2}\,\left (4\,{\mathrm {e}}^{12}\,\ln \relax (2)+24\,{\mathrm {e}}^{12}\,{\ln \relax (2)}^2+48\,{\mathrm {e}}^{12}\,{\ln \relax (2)}^3+32\,{\mathrm {e}}^{12}\,{\ln \relax (2)}^4\right )}{8\,{\ln \relax (2)}^4}+\frac {x^2\,{\mathrm {e}}^{x^2}\,\left (\frac {{\mathrm {e}}^8}{2}+4\,{\mathrm {e}}^8\,\ln \relax (2)+12\,{\mathrm {e}}^8\,{\ln \relax (2)}^2+16\,{\mathrm {e}}^8\,{\ln \relax (2)}^3+8\,{\mathrm {e}}^8\,{\ln \relax (2)}^4\right )}{8\,{\ln \relax (2)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*(4*log(2)^2*(exp(8)*(12*x^4 + 12*x^6) + exp(12)*(12*x^3 + 24*x^5) + 12*x^4*exp(16)) + 2*log(2)*(
exp(12)*(4*x^3 + 8*x^5) + exp(8)*(8*x^4 + 8*x^6)) + 8*log(2)^3*(exp(8)*(8*x^4 + 8*x^6) - exp(20)*(4*x - 8*x^3)
+ exp(12)*(12*x^3 + 24*x^5) + 24*x^4*exp(16)) + exp(8)*(2*x^4 + 2*x^6) + 16*log(2)^4*(exp(24)*(2*x^2 - 2) - e
xp(20)*(4*x - 8*x^3) + exp(8)*(2*x^4 + 2*x^6) + exp(12)*(4*x^3 + 8*x^5) + 12*x^4*exp(16))))/(16*x^3*log(2)^4),
x)

[Out]

(exp(x^2)*(3*exp(16) + 12*exp(16)*log(2) + 12*exp(16)*log(2)^2))/(2*log(2)^2) + (exp(x^2 + 24)*log(2)^4 + 2*x*
exp(x^2 + 20)*log(2)^3*(2*log(2) + 1))/(x^2*log(2)^4) + (x*exp(x^2)*(4*exp(12)*log(2) + 24*exp(12)*log(2)^2 +
48*exp(12)*log(2)^3 + 32*exp(12)*log(2)^4))/(8*log(2)^4) + (x^2*exp(x^2)*(exp(8)/2 + 4*exp(8)*log(2) + 12*exp(
8)*log(2)^2 + 16*exp(8)*log(2)^3 + 8*exp(8)*log(2)^4))/(8*log(2)^4)

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sympy [B]  time = 0.39, size = 201, normalized size = 8.38 \begin {gather*} \frac {\left (x^{4} e^{8} + 16 x^{4} e^{8} \log {\relax (2 )}^{4} + 8 x^{4} e^{8} \log {\relax (2 )} + 32 x^{4} e^{8} \log {\relax (2 )}^{3} + 24 x^{4} e^{8} \log {\relax (2 )}^{2} + 8 x^{3} e^{12} \log {\relax (2 )} + 64 x^{3} e^{12} \log {\relax (2 )}^{4} + 48 x^{3} e^{12} \log {\relax (2 )}^{2} + 96 x^{3} e^{12} \log {\relax (2 )}^{3} + 24 x^{2} e^{16} \log {\relax (2 )}^{2} + 96 x^{2} e^{16} \log {\relax (2 )}^{4} + 96 x^{2} e^{16} \log {\relax (2 )}^{3} + 32 x e^{20} \log {\relax (2 )}^{3} + 64 x e^{20} \log {\relax (2 )}^{4} + 16 e^{24} \log {\relax (2 )}^{4}\right ) e^{x^{2}}}{16 x^{2} \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*((2*x**2-2)*exp(2)**4*exp(4)**4+(8*x**3-4*x)*exp(2)**4*exp(4)**3+12*x**4*exp(2)**4*exp(4)**
2+(8*x**5+4*x**3)*exp(2)**4*exp(4)+(2*x**6+2*x**4)*exp(2)**4)*ln(2)**4+8*((8*x**3-4*x)*exp(2)**4*exp(4)**3+24*
x**4*exp(2)**4*exp(4)**2+(24*x**5+12*x**3)*exp(2)**4*exp(4)+(8*x**6+8*x**4)*exp(2)**4)*ln(2)**3+4*(12*x**4*exp
(2)**4*exp(4)**2+(24*x**5+12*x**3)*exp(2)**4*exp(4)+(12*x**6+12*x**4)*exp(2)**4)*ln(2)**2+2*((8*x**5+4*x**3)*e
xp(2)**4*exp(4)+(8*x**6+8*x**4)*exp(2)**4)*ln(2)+(2*x**6+2*x**4)*exp(2)**4)*exp(x**2)/x**3/ln(2)**4,x)

[Out]

(x**4*exp(8) + 16*x**4*exp(8)*log(2)**4 + 8*x**4*exp(8)*log(2) + 32*x**4*exp(8)*log(2)**3 + 24*x**4*exp(8)*log
(2)**2 + 8*x**3*exp(12)*log(2) + 64*x**3*exp(12)*log(2)**4 + 48*x**3*exp(12)*log(2)**2 + 96*x**3*exp(12)*log(2
)**3 + 24*x**2*exp(16)*log(2)**2 + 96*x**2*exp(16)*log(2)**4 + 96*x**2*exp(16)*log(2)**3 + 32*x*exp(20)*log(2)
**3 + 64*x*exp(20)*log(2)**4 + 16*exp(24)*log(2)**4)*exp(x**2)/(16*x**2*log(2)**4)

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