3.71.52 \(\int (2 x+e^{e^{20}+x^4-4 x^5+6 x^6-4 x^7+x^8+e^{15} (4 x-4 x^2)+e^{10} (6 x^2-12 x^3+6 x^4)+e^5 (4 x^3-12 x^4+12 x^5-4 x^6)} (2+16 x^3-72 x^4+104 x^5-40 x^6-24 x^7+16 x^8+e^{15} (16-24 x-16 x^2)+e^{10} (48 x-120 x^2+24 x^3+48 x^4)+e^5 (48 x^2-168 x^3+144 x^4+24 x^5-48 x^6))+\log (4)) \, dx\)

Optimal. Leaf size=28 \[ x^2+e^{\left (e^5+x-x^2\right )^4} (4+2 x)+x \log (4) \]

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Rubi [B]  time = 4.99, antiderivative size = 262, normalized size of antiderivative = 9.36, number of steps used = 2, number of rules used = 1, integrand size = 191, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2288} \begin {gather*} \frac {2 \left (2 x^8-3 x^7-5 x^6+13 x^5-9 x^4+2 x^3+e^{15} \left (-2 x^2-3 x+2\right )+3 e^{10} \left (2 x^4+x^3-5 x^2+2 x\right )+3 e^5 \left (-2 x^6+x^5+6 x^4-7 x^3+2 x^2\right )\right ) \exp \left (x^8-4 x^7+6 x^6-4 x^5+x^4+4 e^{15} \left (x-x^2\right )+6 e^{10} \left (x^4-2 x^3+x^2\right )+4 e^5 \left (-x^6+3 x^5-3 x^4+x^3\right )+e^{20}\right )}{2 x^7-7 x^6+9 x^5-5 x^4+x^3+3 e^{10} \left (2 x^3-3 x^2+x\right )+3 e^5 \left (-2 x^5+5 x^4-4 x^3+x^2\right )+e^{15} (1-2 x)}+x^2+x \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + E^(E^20 + x^4 - 4*x^5 + 6*x^6 - 4*x^7 + x^8 + E^15*(4*x - 4*x^2) + E^10*(6*x^2 - 12*x^3 + 6*x^4) + E
^5*(4*x^3 - 12*x^4 + 12*x^5 - 4*x^6))*(2 + 16*x^3 - 72*x^4 + 104*x^5 - 40*x^6 - 24*x^7 + 16*x^8 + E^15*(16 - 2
4*x - 16*x^2) + E^10*(48*x - 120*x^2 + 24*x^3 + 48*x^4) + E^5*(48*x^2 - 168*x^3 + 144*x^4 + 24*x^5 - 48*x^6))
+ Log[4],x]

[Out]

x^2 + (2*E^(E^20 + x^4 - 4*x^5 + 6*x^6 - 4*x^7 + x^8 + 4*E^15*(x - x^2) + 6*E^10*(x^2 - 2*x^3 + x^4) + 4*E^5*(
x^3 - 3*x^4 + 3*x^5 - x^6))*(2*x^3 - 9*x^4 + 13*x^5 - 5*x^6 - 3*x^7 + 2*x^8 + E^15*(2 - 3*x - 2*x^2) + 3*E^10*
(2*x - 5*x^2 + x^3 + 2*x^4) + 3*E^5*(2*x^2 - 7*x^3 + 6*x^4 + x^5 - 2*x^6)))/(E^15*(1 - 2*x) + x^3 - 5*x^4 + 9*
x^5 - 7*x^6 + 2*x^7 + 3*E^10*(x - 3*x^2 + 2*x^3) + 3*E^5*(x^2 - 4*x^3 + 5*x^4 - 2*x^5)) + x*Log[4]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2+x \log (4)+\int \exp \left (e^{20}+x^4-4 x^5+6 x^6-4 x^7+x^8+e^{15} \left (4 x-4 x^2\right )+e^{10} \left (6 x^2-12 x^3+6 x^4\right )+e^5 \left (4 x^3-12 x^4+12 x^5-4 x^6\right )\right ) \left (2+16 x^3-72 x^4+104 x^5-40 x^6-24 x^7+16 x^8+e^{15} \left (16-24 x-16 x^2\right )+e^{10} \left (48 x-120 x^2+24 x^3+48 x^4\right )+e^5 \left (48 x^2-168 x^3+144 x^4+24 x^5-48 x^6\right )\right ) \, dx\\ &=x^2+\frac {2 \exp \left (e^{20}+x^4-4 x^5+6 x^6-4 x^7+x^8+4 e^{15} \left (x-x^2\right )+6 e^{10} \left (x^2-2 x^3+x^4\right )+4 e^5 \left (x^3-3 x^4+3 x^5-x^6\right )\right ) \left (2 x^3-9 x^4+13 x^5-5 x^6-3 x^7+2 x^8+e^{15} \left (2-3 x-2 x^2\right )+3 e^{10} \left (2 x-5 x^2+x^3+2 x^4\right )+3 e^5 \left (2 x^2-7 x^3+6 x^4+x^5-2 x^6\right )\right )}{e^{15} (1-2 x)+x^3-5 x^4+9 x^5-7 x^6+2 x^7+3 e^{10} \left (x-3 x^2+2 x^3\right )+3 e^5 \left (x^2-4 x^3+5 x^4-2 x^5\right )}+x \log (4)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 26, normalized size = 0.93 \begin {gather*} 2 e^{\left (e^5+x-x^2\right )^4} (2+x)+x (x+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + E^(E^20 + x^4 - 4*x^5 + 6*x^6 - 4*x^7 + x^8 + E^15*(4*x - 4*x^2) + E^10*(6*x^2 - 12*x^3 + 6*x^
4) + E^5*(4*x^3 - 12*x^4 + 12*x^5 - 4*x^6))*(2 + 16*x^3 - 72*x^4 + 104*x^5 - 40*x^6 - 24*x^7 + 16*x^8 + E^15*(
16 - 24*x - 16*x^2) + E^10*(48*x - 120*x^2 + 24*x^3 + 48*x^4) + E^5*(48*x^2 - 168*x^3 + 144*x^4 + 24*x^5 - 48*
x^6)) + Log[4],x]

[Out]

2*E^(E^5 + x - x^2)^4*(2 + x) + x*(x + Log[4])

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fricas [B]  time = 0.72, size = 89, normalized size = 3.18 \begin {gather*} x^{2} + 2 \, {\left (x + 2\right )} e^{\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4} - 4 \, {\left (x^{2} - x\right )} e^{15} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{10} - 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{5} + e^{20}\right )} + 2 \, x \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-24*x+16)*exp(5)^3+(48*x^4+24*x^3-120*x^2+48*x)*exp(5)^2+(-48*x^6+24*x^5+144*x^4-168*x^3+48
*x^2)*exp(5)+16*x^8-24*x^7-40*x^6+104*x^5-72*x^4+16*x^3+2)*exp(exp(5)^4+(-4*x^2+4*x)*exp(5)^3+(6*x^4-12*x^3+6*
x^2)*exp(5)^2+(-4*x^6+12*x^5-12*x^4+4*x^3)*exp(5)+x^8-4*x^7+6*x^6-4*x^5+x^4)+2*log(2)+2*x,x, algorithm="fricas
")

[Out]

x^2 + 2*(x + 2)*e^(x^8 - 4*x^7 + 6*x^6 - 4*x^5 + x^4 - 4*(x^2 - x)*e^15 + 6*(x^4 - 2*x^3 + x^2)*e^10 - 4*(x^6
- 3*x^5 + 3*x^4 - x^3)*e^5 + e^20) + 2*x*log(2)

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giac [B]  time = 0.71, size = 192, normalized size = 6.86 \begin {gather*} x^{2} + 2 \, {\left (x e^{\left (x^{8} - 4 \, x^{7} - 4 \, x^{6} e^{5} + 6 \, x^{6} + 12 \, x^{5} e^{5} - 4 \, x^{5} + 6 \, x^{4} e^{10} - 12 \, x^{4} e^{5} + x^{4} - 12 \, x^{3} e^{10} + 4 \, x^{3} e^{5} - 4 \, x^{2} e^{15} + 6 \, x^{2} e^{10} + 4 \, x e^{15} + e^{20} + 5\right )} + 2 \, e^{\left (x^{8} - 4 \, x^{7} - 4 \, x^{6} e^{5} + 6 \, x^{6} + 12 \, x^{5} e^{5} - 4 \, x^{5} + 6 \, x^{4} e^{10} - 12 \, x^{4} e^{5} + x^{4} - 12 \, x^{3} e^{10} + 4 \, x^{3} e^{5} - 4 \, x^{2} e^{15} + 6 \, x^{2} e^{10} + 4 \, x e^{15} + e^{20} + 5\right )}\right )} e^{\left (-5\right )} + 2 \, x \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-24*x+16)*exp(5)^3+(48*x^4+24*x^3-120*x^2+48*x)*exp(5)^2+(-48*x^6+24*x^5+144*x^4-168*x^3+48
*x^2)*exp(5)+16*x^8-24*x^7-40*x^6+104*x^5-72*x^4+16*x^3+2)*exp(exp(5)^4+(-4*x^2+4*x)*exp(5)^3+(6*x^4-12*x^3+6*
x^2)*exp(5)^2+(-4*x^6+12*x^5-12*x^4+4*x^3)*exp(5)+x^8-4*x^7+6*x^6-4*x^5+x^4)+2*log(2)+2*x,x, algorithm="giac")

[Out]

x^2 + 2*(x*e^(x^8 - 4*x^7 - 4*x^6*e^5 + 6*x^6 + 12*x^5*e^5 - 4*x^5 + 6*x^4*e^10 - 12*x^4*e^5 + x^4 - 12*x^3*e^
10 + 4*x^3*e^5 - 4*x^2*e^15 + 6*x^2*e^10 + 4*x*e^15 + e^20 + 5) + 2*e^(x^8 - 4*x^7 - 4*x^6*e^5 + 6*x^6 + 12*x^
5*e^5 - 4*x^5 + 6*x^4*e^10 - 12*x^4*e^5 + x^4 - 12*x^3*e^10 + 4*x^3*e^5 - 4*x^2*e^15 + 6*x^2*e^10 + 4*x*e^15 +
e^20 + 5))*e^(-5) + 2*x*log(2)

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maple [B]  time = 37.44, size = 102, normalized size = 3.64

method	result	size
risch	\(\left (2 x +4\right ) {\mathrm e}^{x^{8}-4 x^{6} {\mathrm e}^{5}-4 x^{7}+12 x^{5} {\mathrm e}^{5}+6 x^{6}+6 x^{4} {\mathrm e}^{10}-12 x^{4} {\mathrm e}^{5}-4 x^{5}-12 x^{3} {\mathrm e}^{10}+4 x^{3} {\mathrm e}^{5}+x^{4}-4 x^{2} {\mathrm e}^{15}+6 x^{2} {\mathrm e}^{10}+4 x \,{\mathrm e}^{15}+{\mathrm e}^{20}}+2 x \ln \relax (2)+x^{2}\)	\(102\)
default	\(2 x \,{\mathrm e}^{{\mathrm e}^{20}+\left (-4 x^{2}+4 x \right ) {\mathrm e}^{15}+\left (6 x^{4}-12 x^{3}+6 x^{2}\right ) {\mathrm e}^{10}+\left (-4 x^{6}+12 x^{5}-12 x^{4}+4 x^{3}\right ) {\mathrm e}^{5}+x^{8}-4 x^{7}+6 x^{6}-4 x^{5}+x^{4}}+4 \,{\mathrm e}^{{\mathrm e}^{20}+\left (-4 x^{2}+4 x \right ) {\mathrm e}^{15}+\left (6 x^{4}-12 x^{3}+6 x^{2}\right ) {\mathrm e}^{10}+\left (-4 x^{6}+12 x^{5}-12 x^{4}+4 x^{3}\right ) {\mathrm e}^{5}+x^{8}-4 x^{7}+6 x^{6}-4 x^{5}+x^{4}}+x^{2}+2 x \ln \relax (2)\)	\(187\)
norman	\(2 x \,{\mathrm e}^{{\mathrm e}^{20}+\left (-4 x^{2}+4 x \right ) {\mathrm e}^{15}+\left (6 x^{4}-12 x^{3}+6 x^{2}\right ) {\mathrm e}^{10}+\left (-4 x^{6}+12 x^{5}-12 x^{4}+4 x^{3}\right ) {\mathrm e}^{5}+x^{8}-4 x^{7}+6 x^{6}-4 x^{5}+x^{4}}+4 \,{\mathrm e}^{{\mathrm e}^{20}+\left (-4 x^{2}+4 x \right ) {\mathrm e}^{15}+\left (6 x^{4}-12 x^{3}+6 x^{2}\right ) {\mathrm e}^{10}+\left (-4 x^{6}+12 x^{5}-12 x^{4}+4 x^{3}\right ) {\mathrm e}^{5}+x^{8}-4 x^{7}+6 x^{6}-4 x^{5}+x^{4}}+x^{2}+2 x \ln \relax (2)\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^2-24*x+16)*exp(5)^3+(48*x^4+24*x^3-120*x^2+48*x)*exp(5)^2+(-48*x^6+24*x^5+144*x^4-168*x^3+48*x^2)*
exp(5)+16*x^8-24*x^7-40*x^6+104*x^5-72*x^4+16*x^3+2)*exp(exp(5)^4+(-4*x^2+4*x)*exp(5)^3+(6*x^4-12*x^3+6*x^2)*e
xp(5)^2+(-4*x^6+12*x^5-12*x^4+4*x^3)*exp(5)+x^8-4*x^7+6*x^6-4*x^5+x^4)+2*ln(2)+2*x,x,method=_RETURNVERBOSE)

[Out]

(2*x+4)*exp(x^8-4*x^6*exp(5)-4*x^7+12*x^5*exp(5)+6*x^6+6*x^4*exp(10)-12*x^4*exp(5)-4*x^5-12*x^3*exp(10)+4*x^3*
exp(5)+x^4-4*x^2*exp(15)+6*x^2*exp(10)+4*x*exp(15)+exp(20))+2*x*ln(2)+x^2

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maxima [B]  time = 0.44, size = 106, normalized size = 3.79 \begin {gather*} x^{2} + 2 \, {\left (x e^{\left (e^{20}\right )} + 2 \, e^{\left (e^{20}\right )}\right )} e^{\left (x^{8} - 4 \, x^{7} - 4 \, x^{6} e^{5} + 6 \, x^{6} + 12 \, x^{5} e^{5} - 4 \, x^{5} + 6 \, x^{4} e^{10} - 12 \, x^{4} e^{5} + x^{4} - 12 \, x^{3} e^{10} + 4 \, x^{3} e^{5} - 4 \, x^{2} e^{15} + 6 \, x^{2} e^{10} + 4 \, x e^{15}\right )} + 2 \, x \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2-24*x+16)*exp(5)^3+(48*x^4+24*x^3-120*x^2+48*x)*exp(5)^2+(-48*x^6+24*x^5+144*x^4-168*x^3+48
*x^2)*exp(5)+16*x^8-24*x^7-40*x^6+104*x^5-72*x^4+16*x^3+2)*exp(exp(5)^4+(-4*x^2+4*x)*exp(5)^3+(6*x^4-12*x^3+6*
x^2)*exp(5)^2+(-4*x^6+12*x^5-12*x^4+4*x^3)*exp(5)+x^8-4*x^7+6*x^6-4*x^5+x^4)+2*log(2)+2*x,x, algorithm="maxima
")

[Out]

x^2 + 2*(x*e^(e^20) + 2*e^(e^20))*e^(x^8 - 4*x^7 - 4*x^6*e^5 + 6*x^6 + 12*x^5*e^5 - 4*x^5 + 6*x^4*e^10 - 12*x^
4*e^5 + x^4 - 12*x^3*e^10 + 4*x^3*e^5 - 4*x^2*e^15 + 6*x^2*e^10 + 4*x*e^15) + 2*x*log(2)

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mupad [B]  time = 4.59, size = 186, normalized size = 6.64 \begin {gather*} 4\,{\mathrm {e}}^{{\mathrm {e}}^{20}+4\,x\,{\mathrm {e}}^{15}+4\,x^3\,{\mathrm {e}}^5-12\,x^4\,{\mathrm {e}}^5+12\,x^5\,{\mathrm {e}}^5-4\,x^6\,{\mathrm {e}}^5+6\,x^2\,{\mathrm {e}}^{10}-12\,x^3\,{\mathrm {e}}^{10}+6\,x^4\,{\mathrm {e}}^{10}-4\,x^2\,{\mathrm {e}}^{15}+x^4-4\,x^5+6\,x^6-4\,x^7+x^8}+2\,x\,\ln \relax (2)+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{20}+4\,x\,{\mathrm {e}}^{15}+4\,x^3\,{\mathrm {e}}^5-12\,x^4\,{\mathrm {e}}^5+12\,x^5\,{\mathrm {e}}^5-4\,x^6\,{\mathrm {e}}^5+6\,x^2\,{\mathrm {e}}^{10}-12\,x^3\,{\mathrm {e}}^{10}+6\,x^4\,{\mathrm {e}}^{10}-4\,x^2\,{\mathrm {e}}^{15}+x^4-4\,x^5+6\,x^6-4\,x^7+x^8}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 2*log(2) + exp(exp(20) + exp(15)*(4*x - 4*x^2) + exp(10)*(6*x^2 - 12*x^3 + 6*x^4) + x^4 - 4*x^5 + 6*
x^6 - 4*x^7 + x^8 + exp(5)*(4*x^3 - 12*x^4 + 12*x^5 - 4*x^6))*(exp(5)*(48*x^2 - 168*x^3 + 144*x^4 + 24*x^5 - 4
8*x^6) - exp(15)*(24*x + 16*x^2 - 16) + exp(10)*(48*x - 120*x^2 + 24*x^3 + 48*x^4) + 16*x^3 - 72*x^4 + 104*x^5
- 40*x^6 - 24*x^7 + 16*x^8 + 2),x)

[Out]

4*exp(exp(20) + 4*x*exp(15) + 4*x^3*exp(5) - 12*x^4*exp(5) + 12*x^5*exp(5) - 4*x^6*exp(5) + 6*x^2*exp(10) - 12
*x^3*exp(10) + 6*x^4*exp(10) - 4*x^2*exp(15) + x^4 - 4*x^5 + 6*x^6 - 4*x^7 + x^8) + 2*x*log(2) + 2*x*exp(exp(2
0) + 4*x*exp(15) + 4*x^3*exp(5) - 12*x^4*exp(5) + 12*x^5*exp(5) - 4*x^6*exp(5) + 6*x^2*exp(10) - 12*x^3*exp(10
) + 6*x^4*exp(10) - 4*x^2*exp(15) + x^4 - 4*x^5 + 6*x^6 - 4*x^7 + x^8) + x^2

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sympy [B]  time = 0.44, size = 95, normalized size = 3.39 \begin {gather*} x^{2} + 2 x \log {\relax (2 )} + \left (2 x + 4\right ) e^{x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + x^{4} + \left (- 4 x^{2} + 4 x\right ) e^{15} + \left (6 x^{4} - 12 x^{3} + 6 x^{2}\right ) e^{10} + \left (- 4 x^{6} + 12 x^{5} - 12 x^{4} + 4 x^{3}\right ) e^{5} + e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**2-24*x+16)*exp(5)**3+(48*x**4+24*x**3-120*x**2+48*x)*exp(5)**2+(-48*x**6+24*x**5+144*x**4-1
68*x**3+48*x**2)*exp(5)+16*x**8-24*x**7-40*x**6+104*x**5-72*x**4+16*x**3+2)*exp(exp(5)**4+(-4*x**2+4*x)*exp(5)
**3+(6*x**4-12*x**3+6*x**2)*exp(5)**2+(-4*x**6+12*x**5-12*x**4+4*x**3)*exp(5)+x**8-4*x**7+6*x**6-4*x**5+x**4)+
2*ln(2)+2*x,x)

[Out]

x**2 + 2*x*log(2) + (2*x + 4)*exp(x**8 - 4*x**7 + 6*x**6 - 4*x**5 + x**4 + (-4*x**2 + 4*x)*exp(15) + (6*x**4 -
12*x**3 + 6*x**2)*exp(10) + (-4*x**6 + 12*x**5 - 12*x**4 + 4*x**3)*exp(5) + exp(20))

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