3.10.31 \(\int \frac {-e^5+e^{\frac {4+e^5 (12-3 x)-13 x+3 x^2+(4-x) \log (3)}{e^5}} (-13-3 e^5+6 x-\log (3))}{e^5} \, dx\)

Optimal. Leaf size=28 \[ e^{\frac {(-4+x) \left (-1+3 \left (-e^5+x\right )-\log (3)\right )}{e^5}}-x \]

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Rubi [A]  time = 0.28, antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 3, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {12, 2244, 2236} \begin {gather*} \exp \left (\frac {3 x^2}{e^5}-\frac {x \left (13+3 e^5+\log (3)\right )}{e^5}+\frac {4+12 e^5+\log (81)}{e^5}\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^5 + E^((4 + E^5*(12 - 3*x) - 13*x + 3*x^2 + (4 - x)*Log[3])/E^5)*(-13 - 3*E^5 + 6*x - Log[3]))/E^5,x]

[Out]

E^((3*x^2)/E^5 - (x*(13 + 3*E^5 + Log[3]))/E^5 + (4 + 12*E^5 + Log[81])/E^5) - x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-e^5+\exp \left (\frac {4+e^5 (12-3 x)-13 x+3 x^2+(4-x) \log (3)}{e^5}\right ) \left (-13-3 e^5+6 x-\log (3)\right )\right ) \, dx}{e^5}\\ &=-x+\frac {\int \exp \left (\frac {4+e^5 (12-3 x)-13 x+3 x^2+(4-x) \log (3)}{e^5}\right ) \left (-13-3 e^5+6 x-\log (3)\right ) \, dx}{e^5}\\ &=-x+\frac {\int \exp \left (\frac {3 x^2}{e^5}+\frac {4 \left (1+3 e^5+\log (3)\right )}{e^5}-\frac {x \left (13+3 e^5+\log (3)\right )}{e^5}\right ) \left (-13-3 e^5+6 x-\log (3)\right ) \, dx}{e^5}\\ &=\exp \left (\frac {3 x^2}{e^5}-\frac {x \left (13+3 e^5+\log (3)\right )}{e^5}+\frac {4+12 e^5+\log (81)}{e^5}\right )-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 44, normalized size = 1.57 \begin {gather*} e^{5+\frac {3 x^2}{e^5}-\frac {x \left (13+3 e^5+\log (3)\right )}{e^5}+\frac {4+7 e^5+\log (81)}{e^5}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^5 + E^((4 + E^5*(12 - 3*x) - 13*x + 3*x^2 + (4 - x)*Log[3])/E^5)*(-13 - 3*E^5 + 6*x - Log[3]))/E
^5,x]

[Out]

E^(5 + (3*x^2)/E^5 - (x*(13 + 3*E^5 + Log[3]))/E^5 + (4 + 7*E^5 + Log[81])/E^5) - x

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fricas [A]  time = 0.71, size = 32, normalized size = 1.14 \begin {gather*} -x + e^{\left ({\left (3 \, x^{2} - 3 \, {\left (x - 4\right )} e^{5} - {\left (x - 4\right )} \log \relax (3) - 13 \, x + 4\right )} e^{\left (-5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(3)-3*exp(5)+6*x-13)*exp(((-x+4)*log(3)+(-3*x+12)*exp(5)+3*x^2-13*x+4)/exp(5))-exp(5))/exp(5),
x, algorithm="fricas")

[Out]

-x + e^((3*x^2 - 3*(x - 4)*e^5 - (x - 4)*log(3) - 13*x + 4)*e^(-5))

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giac [A]  time = 0.45, size = 46, normalized size = 1.64 \begin {gather*} -{\left (x e^{5} - e^{\left (3 \, x^{2} e^{\left (-5\right )} - x e^{\left (-5\right )} \log \relax (3) - 13 \, x e^{\left (-5\right )} + 4 \, e^{\left (-5\right )} \log \relax (3) - 3 \, x + 4 \, e^{\left (-5\right )} + 17\right )}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(3)-3*exp(5)+6*x-13)*exp(((-x+4)*log(3)+(-3*x+12)*exp(5)+3*x^2-13*x+4)/exp(5))-exp(5))/exp(5),
x, algorithm="giac")

[Out]

-(x*e^5 - e^(3*x^2*e^(-5) - x*e^(-5)*log(3) - 13*x*e^(-5) + 4*e^(-5)*log(3) - 3*x + 4*e^(-5) + 17))*e^(-5)

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maple [A]  time = 0.04, size = 24, normalized size = 0.86

method	result	size
risch	\(-x +{\mathrm e}^{-\left (x -4\right ) \left (-3 x +\ln \relax (3)+3 \,{\mathrm e}^{5}+1\right ) {\mathrm e}^{-5}}\)	\(24\)
norman	\(-x +{\mathrm e}^{\left (\left (-x +4\right ) \ln \relax (3)+\left (-3 x +12\right ) {\mathrm e}^{5}+3 x^{2}-13 x +4\right ) {\mathrm e}^{-5}}\)	\(37\)
default	\({\mathrm e}^{-5} \left ({\mathrm e}^{\left (\left (-x +4\right ) \ln \relax (3)+\left (-3 x +12\right ) {\mathrm e}^{5}+3 x^{2}-13 x +4\right ) {\mathrm e}^{-5}} {\mathrm e}^{5}-x \,{\mathrm e}^{5}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(3)-3*exp(5)+6*x-13)*exp(((-x+4)*ln(3)+(-3*x+12)*exp(5)+3*x^2-13*x+4)/exp(5))-exp(5))/exp(5),x,method
=_RETURNVERBOSE)

[Out]

-x+exp(-(x-4)*(-3*x+ln(3)+3*exp(5)+1)*exp(-5))

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maxima [A]  time = 0.48, size = 41, normalized size = 1.46 \begin {gather*} -{\left (x e^{5} - e^{\left ({\left (3 \, x^{2} - 3 \, {\left (x - 4\right )} e^{5} - {\left (x - 4\right )} \log \relax (3) - 13 \, x + 4\right )} e^{\left (-5\right )} + 5\right )}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(3)-3*exp(5)+6*x-13)*exp(((-x+4)*log(3)+(-3*x+12)*exp(5)+3*x^2-13*x+4)/exp(5))-exp(5))/exp(5),
x, algorithm="maxima")

[Out]

-(x*e^5 - e^((3*x^2 - 3*(x - 4)*e^5 - (x - 4)*log(3) - 13*x + 4)*e^(-5) + 5))*e^(-5)

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mupad [B]  time = 0.39, size = 44, normalized size = 1.57 \begin {gather*} \frac {3^{4\,{\mathrm {e}}^{-5}}\,{\mathrm {e}}^{3\,x^2\,{\mathrm {e}}^{-5}}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-5}}\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{12}\,{\mathrm {e}}^{-13\,x\,{\mathrm {e}}^{-5}}}{3^{x\,{\mathrm {e}}^{-5}}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-5)*(exp(5) + exp(-exp(-5)*(13*x + log(3)*(x - 4) - 3*x^2 + exp(5)*(3*x - 12) - 4))*(3*exp(5) - 6*x +
log(3) + 13)),x)

[Out]

(3^(4*exp(-5))*exp(3*x^2*exp(-5))*exp(4*exp(-5))*exp(-3*x)*exp(12)*exp(-13*x*exp(-5)))/3^(x*exp(-5)) - x

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sympy [A]  time = 0.15, size = 31, normalized size = 1.11 \begin {gather*} - x + e^{\frac {3 x^{2} - 13 x + \left (4 - x\right ) \log {\relax (3 )} + \left (12 - 3 x\right ) e^{5} + 4}{e^{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(3)-3*exp(5)+6*x-13)*exp(((-x+4)*ln(3)+(-3*x+12)*exp(5)+3*x**2-13*x+4)/exp(5))-exp(5))/exp(5),x
)

[Out]

-x + exp((3*x**2 - 13*x + (4 - x)*log(3) + (12 - 3*x)*exp(5) + 4)*exp(-5))

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