3.71.86 \(\int (1+e^{-5-4 x+e^5 (20 x-5 x^2)} (4+e^5 (-20+10 x))) \, dx\) [7086]

Optimal. Leaf size=26 \[ 2-e^{\left (-4+5 e^5 (4-x)-\frac {5}{x}\right ) x}+x \]

[Out]

2+x-exp(x*(5*(4-x)*exp(5)-5/x-4))

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Rubi [A]
time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2276, 2268} \begin {gather*} x-e^{-5 e^5 x^2-4 \left (1-5 e^5\right ) x-5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(-5 - 4*x + E^5*(20*x - 5*x^2))*(4 + E^5*(-20 + 10*x)),x]

[Out]

-E^(-5 - 4*(1 - 5*E^5)*x - 5*E^5*x^2) + x

Rule 2268

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 2276

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} &=x+\int e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right ) \, dx\\ &=x+\int e^{-5-4 \left (1-5 e^5\right ) x-5 e^5 x^2} \left (4 \left (1-5 e^5\right )+10 e^5 x\right ) \, dx\\ &=-e^{-5-4 \left (1-5 e^5\right ) x-5 e^5 x^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.96 \begin {gather*} -e^{-5-4 x+20 e^5 x-5 e^5 x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(-5 - 4*x + E^5*(20*x - 5*x^2))*(4 + E^5*(-20 + 10*x)),x]

[Out]

-E^(-5 - 4*x + 20*E^5*x - 5*E^5*x^2) + x

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.13, size = 191, normalized size = 7.35

method	result	size
norman	\(x -{\mathrm e}^{\left (-5 x^{2}+20 x \right ) {\mathrm e}^{5}-4 x -5}\)	\(23\)
risch	\(x -{\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}-4 x -5}\)	\(23\)
default	\(x +\frac {2 \sqrt {\pi }\, {\mathrm e}^{-5+\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{5}-2 \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )-{\mathrm e}^{-5} {\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+\left (20 \,{\mathrm e}^{5}-4\right ) x}+\frac {\left (20 \,{\mathrm e}^{5}-4\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{10}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x,method=_RETURNVERBOSE)

[Out]

x+2/5*Pi^(1/2)*exp(-5+1/20*(20*exp(5)-4)^2/exp(5))*5^(1/2)/exp(5/2)*erf(5^(1/2)*exp(5/2)*x-1/10*(20*exp(5)-4)*
5^(1/2)/exp(5/2))-2*Pi^(1/2)*exp(1/20*(20*exp(5)-4)^2/exp(5))*5^(1/2)/exp(5/2)*erf(5^(1/2)*exp(5/2)*x-1/10*(20
*exp(5)-4)*5^(1/2)/exp(5/2))-1/exp(5)*exp(-5*x^2*exp(5)+(20*exp(5)-4)*x)+1/10*(20*exp(5)-4)/exp(5)*Pi^(1/2)*ex
p(1/20*(20*exp(5)-4)^2/exp(5))*5^(1/2)/exp(5/2)*erf(5^(1/2)*exp(5/2)*x-1/10*(20*exp(5)-4)*5^(1/2)/exp(5/2))

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.54, size = 223, normalized size = 8.58 \begin {gather*} -2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {5}{2}\right )} + \frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {15}{2}\right )} + \frac {\sqrt {5} {\left (\frac {2 \, \sqrt {5} \sqrt {\frac {1}{5}} \sqrt {\pi } {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{5}} \sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} e^{\left (-\frac {5}{2}\right )}\right ) - 1\right )} {\left (5 \, e^{5} - 1\right )} e^{\frac {5}{2}}}{\sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} \left (-e^{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {5} e^{\left (-\frac {1}{5} \, {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2} e^{\left (-5\right )} + 5\right )}}{\left (-e^{5}\right )^{\frac {3}{2}}}\right )} e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )}\right )}}{5 \, \sqrt {-e^{5}}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algorithm="maxima")

[Out]

-2*sqrt(5)*sqrt(pi)*erf(sqrt(5)*x*e^(5/2) - 2/5*sqrt(5)*(5*e^5 - 1)*e^(-5/2))*e^(4/5*(5*e^5 - 1)^2*e^(-5) - 5/
2) + 2/5*sqrt(5)*sqrt(pi)*erf(sqrt(5)*x*e^(5/2) - 2/5*sqrt(5)*(5*e^5 - 1)*e^(-5/2))*e^(4/5*(5*e^5 - 1)^2*e^(-5
) - 15/2) + 1/5*sqrt(5)*(2*sqrt(5)*sqrt(1/5)*sqrt(pi)*(5*x*e^5 - 10*e^5 + 2)*(erf(sqrt(1/5)*sqrt((5*x*e^5 - 10
*e^5 + 2)^2)*e^(-5/2)) - 1)*(5*e^5 - 1)*e^(5/2)/(sqrt((5*x*e^5 - 10*e^5 + 2)^2)*(-e^5)^(3/2)) - sqrt(5)*e^(-1/
5*(5*x*e^5 - 10*e^5 + 2)^2*e^(-5) + 5)/(-e^5)^(3/2))*e^(4/5*(5*e^5 - 1)^2*e^(-5))/sqrt(-e^5) + x

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Fricas [A]
time = 0.40, size = 21, normalized size = 0.81 \begin {gather*} x - e^{\left (-5 \, {\left (x^{2} - 4 \, x\right )} e^{5} - 4 \, x - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algorithm="fricas")

[Out]

x - e^(-5*(x^2 - 4*x)*e^5 - 4*x - 5)

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Sympy [A]
time = 0.06, size = 19, normalized size = 0.73 \begin {gather*} x - e^{- 4 x + \left (- 5 x^{2} + 20 x\right ) e^{5} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-20)*exp(5)+4)*exp((-5*x**2+20*x)*exp(5)-4*x-5)+1,x)

[Out]

x - exp(-4*x + (-5*x**2 + 20*x)*exp(5) - 5)

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Giac [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.42, size = 120, normalized size = 4.62 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {1}{5} \, {\left (100 \, e^{10} - 65 \, e^{5} + 4\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} - \frac {1}{5} \, {\left (2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {4}{5} \, {\left (25 \, e^{10} - 10 \, e^{5} + 1\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} + 5 \, e^{\left (-5 \, x^{2} e^{5} + 20 \, x e^{5} - 4 \, x\right )}\right )} e^{\left (-5\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algorithm="giac")

[Out]

2/5*sqrt(5)*sqrt(pi)*erf(-1/5*sqrt(5)*(2*(5*e^5 - 1)*e^(-5) - 5*x)*e^(5/2))*e^(1/5*(100*e^10 - 65*e^5 + 4)*e^(
-5) - 5/2) - 1/5*(2*sqrt(5)*sqrt(pi)*erf(-1/5*sqrt(5)*(2*(5*e^5 - 1)*e^(-5) - 5*x)*e^(5/2))*e^(4/5*(25*e^10 -
10*e^5 + 1)*e^(-5) - 5/2) + 5*e^(-5*x^2*e^5 + 20*x*e^5 - 4*x))*e^(-5) + x

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Mupad [B]
time = 4.30, size = 24, normalized size = 0.92 \begin {gather*} x-{\mathrm {e}}^{-5\,x^2\,{\mathrm {e}}^5}\,{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{20\,x\,{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(5)*(20*x - 5*x^2) - 4*x - 5)*(exp(5)*(10*x - 20) + 4) + 1,x)

[Out]

x - exp(-5*x^2*exp(5))*exp(-4*x)*exp(-5)*exp(20*x*exp(5))

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