3.48.67 \(\int (2+e^{-x^2+x^3+(-x+x^2) \log (3)+(-x-\log (3)) \log (4)} (2 x-3 x^2+(1-2 x) \log (3)+\log (4))) \, dx\)

Optimal. Leaf size=25 \[ -1-e^{(x+\log (3)) (x+(-2+x) x-\log (4))}+2 x \]

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Rubi [A]  time = 0.30, antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 1, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6706} \begin {gather*} 2 x-3^{x^2-x} e^{x^3-x^2} 4^{-x-\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + E^(-x^2 + x^3 + (-x + x^2)*Log[3] + (-x - Log[3])*Log[4])*(2*x - 3*x^2 + (1 - 2*x)*Log[3] + Log[4]),x]

[Out]

-(3^(-x + x^2)*4^(-x - Log[3])*E^(-x^2 + x^3)) + 2*x

Rule 6706

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 {gather*} \begin {aligned} \text {integral} &=2 x+\int \exp \left (-x^2+x^3+\left (-x+x^2\right ) \log (3)+(-x-\log (3)) \log (4)\right ) \left (2 x-3 x^2+(1-2 x) \log (3)+\log (4)\right ) \, dx\\ &=-3^{-x+x^2} 4^{-x-\log (3)} e^{-x^2+x^3}+2 x\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2+e^{-x^2+x^3+\left (-x+x^2\right ) \log (3)+(-x-\log (3)) \log (4)} \left (2 x-3 x^2+(1-2 x) \log (3)+\log (4)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[2 + E^(-x^2 + x^3 + (-x + x^2)*Log[3] + (-x - Log[3])*Log[4])*(2*x - 3*x^2 + (1 - 2*x)*Log[3] + Log[
4]),x]

[Out]

Integrate[2 + E^(-x^2 + x^3 + (-x + x^2)*Log[3] + (-x - Log[3])*Log[4])*(2*x - 3*x^2 + (1 - 2*x)*Log[3] + Log[
4]), x]

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fricas [A]  time = 0.80, size = 35, normalized size = 1.40 \begin {gather*} 2 \, x - e^{\left (x^{3} - x^{2} + {\left (x^{2} - x - 2 \, \log \relax (2)\right )} \log \relax (3) - 2 \, x \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)+(1-2*x)*log(3)-3*x^2+2*x)*exp(2*(-log(3)-x)*log(2)+(x^2-x)*log(3)+x^3-x^2)+2,x, algorithm=
"fricas")

[Out]

2*x - e^(x^3 - x^2 + (x^2 - x - 2*log(2))*log(3) - 2*x*log(2))

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giac [A]  time = 0.18, size = 38, normalized size = 1.52 \begin {gather*} 2 \, x - e^{\left (x^{3} + x^{2} \log \relax (3) - x^{2} - x \log \relax (3) - 2 \, x \log \relax (2) - 2 \, \log \relax (3) \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)+(1-2*x)*log(3)-3*x^2+2*x)*exp(2*(-log(3)-x)*log(2)+(x^2-x)*log(3)+x^3-x^2)+2,x, algorithm=
"giac")

[Out]

2*x - e^(x^3 + x^2*log(3) - x^2 - x*log(3) - 2*x*log(2) - 2*log(3)*log(2))

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maple [A]  time = 0.09, size = 32, normalized size = 1.28

method	result	size
risch	\(2 x -2^{-2 \ln \relax (3)-2 x} 3^{x \left (x -1\right )} {\mathrm e}^{x^{2} \left (x -1\right )}\)	\(32\)
default	\(2 x -{\mathrm e}^{2 \left (-\ln \relax (3)-x \right ) \ln \relax (2)+\left (x^{2}-x \right ) \ln \relax (3)+x^{3}-x^{2}}\)	\(39\)
norman	\(2 x -{\mathrm e}^{2 \left (-\ln \relax (3)-x \right ) \ln \relax (2)+\left (x^{2}-x \right ) \ln \relax (3)+x^{3}-x^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(2)+(1-2*x)*ln(3)-3*x^2+2*x)*exp(2*(-ln(3)-x)*ln(2)+(x^2-x)*ln(3)+x^3-x^2)+2,x,method=_RETURNVERBOSE)

[Out]

2*x-2^(-2*ln(3)-2*x)*3^(x*(x-1))*exp(x^2*(x-1))

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maxima [A]  time = 0.36, size = 34, normalized size = 1.36 \begin {gather*} 2 \, x - e^{\left (x^{3} - x^{2} + {\left (x^{2} - x\right )} \log \relax (3) - 2 \, {\left (x + \log \relax (3)\right )} \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(2)+(1-2*x)*log(3)-3*x^2+2*x)*exp(2*(-log(3)-x)*log(2)+(x^2-x)*log(3)+x^3-x^2)+2,x, algorithm=
"maxima")

[Out]

2*x - e^(x^3 - x^2 + (x^2 - x)*log(3) - 2*(x + log(3))*log(2))

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mupad [B]  time = 0.16, size = 41, normalized size = 1.64 \begin {gather*} 2\,x-\frac {3^{x^2}\,{\mathrm {e}}^{x^3-x^2}}{2^{2\,\ln \relax (3)}\,2^{2\,x}\,3^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^3 - x^2 - 2*log(2)*(x + log(3)) - log(3)*(x - x^2))*(2*x + 2*log(2) - log(3)*(2*x - 1) - 3*x^2) + 2,
x)

[Out]

2*x - (3^(x^2)*exp(x^3 - x^2))/(2^(2*log(3))*2^(2*x)*3^x)

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sympy [A]  time = 0.15, size = 32, normalized size = 1.28 \begin {gather*} 2 x - e^{x^{3} - x^{2} + \left (- 2 x - 2 \log {\relax (3 )}\right ) \log {\relax (2 )} + \left (x^{2} - x\right ) \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(2)+(1-2*x)*ln(3)-3*x**2+2*x)*exp(2*(-ln(3)-x)*ln(2)+(x**2-x)*ln(3)+x**3-x**2)+2,x)

[Out]

2*x - exp(x**3 - x**2 + (-2*x - 2*log(3))*log(2) + (x**2 - x)*log(3))

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