\(\int e^{158-1000080 x+1562750010 x^2} (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x) \, dx\) [1093]

Optimal result

Integrand size = 30, antiderivative size = 17 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=-2 e^{-2+10 (4-12501 x)^2}+x \]

[Out]

x-2/exp(1-5*(4-12501*x)^2)^2

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6874, 2266, 2235, 2272} \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=x-2 e^{1562750010 x^2-1000080 x+158} \]

[In]

Int[E^(158 - 1000080*x + 1562750010*x^2)*(2000160 + E^(-158 + 1000080*x - 1562750010*x^2) - 6251000040*x),x]

[Out]

-2*E^(158 - 1000080*x + 1562750010*x^2) + x

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2272

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] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (1+2000160 e^{158-1000080 x+1562750010 x^2}-6251000040 e^{158-1000080 x+1562750010 x^2} x\right ) \, dx \\ & = x+2000160 \int e^{158-1000080 x+1562750010 x^2} \, dx-6251000040 \int e^{158-1000080 x+1562750010 x^2} x \, dx \\ & = -2 e^{158-1000080 x+1562750010 x^2}+x-2000160 \int e^{158-1000080 x+1562750010 x^2} \, dx+\frac {2000160 \int e^{\frac {(-1000080+3125500020 x)^2}{6251000040}} \, dx}{e^2} \\ & = -2 e^{158-1000080 x+1562750010 x^2}+x-\frac {8 \sqrt {10 \pi } \text {erfi}\left (\sqrt {10} (4-12501 x)\right )}{e^2}-\frac {2000160 \int e^{\frac {(-1000080+3125500020 x)^2}{6251000040}} \, dx}{e^2} \\ & = -2 e^{158-1000080 x+1562750010 x^2}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=-2 e^{2 \left (79-500040 x+781375005 x^2\right )}+x \]

[In]

Integrate[E^(158 - 1000080*x + 1562750010*x^2)*(2000160 + E^(-158 + 1000080*x - 1562750010*x^2) - 6251000040*x
),x]

[Out]

-2*E^(2*(79 - 500040*x + 781375005*x^2)) + x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
risch	\(x -2 \,{\mathrm e}^{1562750010 x^{2}-1000080 x +158}\)	\(16\)
norman	\(\left (-2+x \,{\mathrm e}^{-1562750010 x^{2}+1000080 x -158}\right ) {\mathrm e}^{1562750010 x^{2}-1000080 x +158}\)	\(32\)
parallelrisch	\(\left (-2+x \,{\mathrm e}^{-1562750010 x^{2}+1000080 x -158}\right ) {\mathrm e}^{1562750010 x^{2}-1000080 x +158}\)	\(32\)
default	\(x -8 i {\mathrm e}^{158} \sqrt {\pi }\, {\mathrm e}^{-160} \sqrt {10}\, \operatorname {erf}\left (12501 i \sqrt {10}\, x -4 i \sqrt {10}\right )-6251000040 \,{\mathrm e}^{158} \left (\frac {{\mathrm e}^{1562750010 x^{2}-1000080 x}}{3125500020}-\frac {i \sqrt {\pi }\, {\mathrm e}^{-160} \sqrt {10}\, \operatorname {erf}\left (12501 i \sqrt {10}\, x -4 i \sqrt {10}\right )}{781375005}\right )\)	\(78\)
parts	\(x -8 i {\mathrm e}^{158} \sqrt {\pi }\, {\mathrm e}^{-160} \sqrt {10}\, \operatorname {erf}\left (12501 i \sqrt {10}\, x -4 i \sqrt {10}\right )-6251000040 \,{\mathrm e}^{158} \left (\frac {{\mathrm e}^{1562750010 x^{2}-1000080 x}}{3125500020}-\frac {i \sqrt {\pi }\, {\mathrm e}^{-160} \sqrt {10}\, \operatorname {erf}\left (12501 i \sqrt {10}\, x -4 i \sqrt {10}\right )}{781375005}\right )\)	\(78\)

[In]

int((exp(-781375005*x^2+500040*x-79)^2-6251000040*x+2000160)/exp(-781375005*x^2+500040*x-79)^2,x,method=_RETUR
NVERBOSE)

[Out]

x-2*exp(1562750010*x^2-1000080*x+158)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=x - 2 \, e^{\left (1562750010 \, x^{2} - 1000080 \, x + 158\right )} \]

[In]

integrate((exp(-781375005*x^2+500040*x-79)^2-6251000040*x+2000160)/exp(-781375005*x^2+500040*x-79)^2,x, algori
thm="fricas")

[Out]

x - 2*e^(1562750010*x^2 - 1000080*x + 158)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=x - 2 e^{1562750010 x^{2} - 1000080 x + 158} \]

[In]

integrate((exp(-781375005*x**2+500040*x-79)**2-6251000040*x+2000160)/exp(-781375005*x**2+500040*x-79)**2,x)

[Out]

x - 2*exp(1562750010*x**2 - 1000080*x + 158)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 5.06 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=-8 i \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (12501 i \, \sqrt {10} x - 4 i \, \sqrt {10}\right ) e^{\left (-2\right )} - \frac {1}{5} \, \sqrt {10} {\left (\frac {40 \, \sqrt {\pi } {\left (12501 \, x - 4\right )} {\left (\operatorname {erf}\left (\sqrt {10} \sqrt {-{\left (12501 \, x - 4\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (12501 \, x - 4\right )}^{2}}} + \sqrt {10} e^{\left (10 \, {\left (12501 \, x - 4\right )}^{2}\right )}\right )} e^{\left (-2\right )} + x \]

[In]

integrate((exp(-781375005*x^2+500040*x-79)^2-6251000040*x+2000160)/exp(-781375005*x^2+500040*x-79)^2,x, algori
thm="maxima")

[Out]

-8*I*sqrt(10)*sqrt(pi)*erf(12501*I*sqrt(10)*x - 4*I*sqrt(10))*e^(-2) - 1/5*sqrt(10)*(40*sqrt(pi)*(12501*x - 4)
*(erf(sqrt(10)*sqrt(-(12501*x - 4)^2)) - 1)/sqrt(-(12501*x - 4)^2) + sqrt(10)*e^(10*(12501*x - 4)^2))*e^(-2) +
x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=x - 2 \, e^{\left (1562750010 \, x^{2} - 1000080 \, x + 158\right )} \]

[In]

integrate((exp(-781375005*x^2+500040*x-79)^2-6251000040*x+2000160)/exp(-781375005*x^2+500040*x-79)^2,x, algori
thm="giac")

[Out]

x - 2*e^(1562750010*x^2 - 1000080*x + 158)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{158-1000080 x+1562750010 x^2} \left (2000160+e^{-158+1000080 x-1562750010 x^2}-6251000040 x\right ) \, dx=x-2\,{\mathrm {e}}^{-1000080\,x}\,{\mathrm {e}}^{158}\,{\mathrm {e}}^{1562750010\,x^2} \]

[In]

int(exp(1562750010*x^2 - 1000080*x + 158)*(exp(1000080*x - 1562750010*x^2 - 158) - 6251000040*x + 2000160),x)

[Out]

x - 2*exp(-1000080*x)*exp(158)*exp(1562750010*x^2)