3.48.49 \(\int (2 x+e^{e^{-x+x^2+8 x^7+16 x^{12}}-x} (1-x+e^{-x+x^2+8 x^7+16 x^{12}} (-x+2 x^2+56 x^7+192 x^{12}))) \, dx\)

Optimal. Leaf size=27 \[ -4+x \left (e^{e^{-x+\left (x+4 x^6\right )^2}-x}+x\right ) \]

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Rubi [B]  time = 0.37, antiderivative size = 108, normalized size of antiderivative = 4.00, number of steps used = 2, number of rules used = 1, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2288} \begin {gather*} x^2+\frac {e^{e^{16 x^{12}+8 x^7+x^2-x}-x} \left (e^{16 x^{12}+8 x^7+x^2-x} \left (-192 x^{12}-56 x^7-2 x^2+x\right )+x\right )}{e^{16 x^{12}+8 x^7+x^2-x} \left (-192 x^{11}-56 x^6-2 x+1\right )+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + E^(E^(-x + x^2 + 8*x^7 + 16*x^12) - x)*(1 - x + E^(-x + x^2 + 8*x^7 + 16*x^12)*(-x + 2*x^2 + 56*x^7
+ 192*x^12)),x]

[Out]

x^2 + (E^(E^(-x + x^2 + 8*x^7 + 16*x^12) - x)*(x + E^(-x + x^2 + 8*x^7 + 16*x^12)*(x - 2*x^2 - 56*x^7 - 192*x^
12)))/(1 + E^(-x + x^2 + 8*x^7 + 16*x^12)*(1 - 2*x - 56*x^6 - 192*x^11))

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+\int e^{e^{-x+x^2+8 x^7+16 x^{12}}-x} \left (1-x+e^{-x+x^2+8 x^7+16 x^{12}} \left (-x+2 x^2+56 x^7+192 x^{12}\right )\right ) \, dx\\ &=x^2+\frac {e^{e^{-x+x^2+8 x^7+16 x^{12}}-x} \left (x+e^{-x+x^2+8 x^7+16 x^{12}} \left (x-2 x^2-56 x^7-192 x^{12}\right )\right )}{1+e^{-x+x^2+8 x^7+16 x^{12}} \left (1-2 x-56 x^6-192 x^{11}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 25, normalized size = 0.93 \begin {gather*} x \left (e^{e^{-x+\left (x+4 x^6\right )^2}-x}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + E^(E^(-x + x^2 + 8*x^7 + 16*x^12) - x)*(1 - x + E^(-x + x^2 + 8*x^7 + 16*x^12)*(-x + 2*x^2 + 5
6*x^7 + 192*x^12)),x]

[Out]

x*(E^(E^(-x + (x + 4*x^6)^2) - x) + x)

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fricas [A]  time = 0.60, size = 29, normalized size = 1.07 \begin {gather*} x^{2} + x e^{\left (-x + e^{\left (16 \, x^{12} + 8 \, x^{7} + x^{2} - x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^12+56*x^7+2*x^2-x)*exp(16*x^12+8*x^7+x^2-x)-x+1)*exp(exp(16*x^12+8*x^7+x^2-x)-x)+2*x,x, algo
rithm="fricas")

[Out]

x^2 + x*e^(-x + e^(16*x^12 + 8*x^7 + x^2 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left ({\left (192 \, x^{12} + 56 \, x^{7} + 2 \, x^{2} - x\right )} e^{\left (16 \, x^{12} + 8 \, x^{7} + x^{2} - x\right )} - x + 1\right )} e^{\left (-x + e^{\left (16 \, x^{12} + 8 \, x^{7} + x^{2} - x\right )}\right )} + 2 \, x\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^12+56*x^7+2*x^2-x)*exp(16*x^12+8*x^7+x^2-x)-x+1)*exp(exp(16*x^12+8*x^7+x^2-x)-x)+2*x,x, algo
rithm="giac")

[Out]

integrate(((192*x^12 + 56*x^7 + 2*x^2 - x)*e^(16*x^12 + 8*x^7 + x^2 - x) - x + 1)*e^(-x + e^(16*x^12 + 8*x^7 +
x^2 - x)) + 2*x, x)

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maple [A]  time = 0.07, size = 28, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{x \left (16 x^{11}+8 x^{6}+x -1\right )}-x} x +x^{2}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((192*x^12+56*x^7+2*x^2-x)*exp(16*x^12+8*x^7+x^2-x)-x+1)*exp(exp(16*x^12+8*x^7+x^2-x)-x)+2*x,x,method=_RET
URNVERBOSE)

[Out]

exp(exp(x*(16*x^11+8*x^6+x-1))-x)*x+x^2

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maxima [A]  time = 0.51, size = 29, normalized size = 1.07 \begin {gather*} x^{2} + x e^{\left (-x + e^{\left (16 \, x^{12} + 8 \, x^{7} + x^{2} - x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^12+56*x^7+2*x^2-x)*exp(16*x^12+8*x^7+x^2-x)-x+1)*exp(exp(16*x^12+8*x^7+x^2-x)-x)+2*x,x, algo
rithm="maxima")

[Out]

x^2 + x*e^(-x + e^(16*x^12 + 8*x^7 + x^2 - x))

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mupad [B]  time = 3.34, size = 32, normalized size = 1.19 \begin {gather*} x^2+x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{8\,x^7}\,{\mathrm {e}}^{16\,x^{12}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + exp(exp(x^2 - x + 8*x^7 + 16*x^12) - x)*(exp(x^2 - x + 8*x^7 + 16*x^12)*(2*x^2 - x + 56*x^7 + 192*x^
12) - x + 1),x)

[Out]

x^2 + x*exp(-x)*exp(exp(-x)*exp(x^2)*exp(8*x^7)*exp(16*x^12))

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sympy [A]  time = 3.88, size = 24, normalized size = 0.89 \begin {gather*} x^{2} + x e^{- x + e^{16 x^{12} + 8 x^{7} + x^{2} - x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x**12+56*x**7+2*x**2-x)*exp(16*x**12+8*x**7+x**2-x)-x+1)*exp(exp(16*x**12+8*x**7+x**2-x)-x)+2*
x,x)

[Out]

x**2 + x*exp(-x + exp(16*x**12 + 8*x**7 + x**2 - x))

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