3.45.7 \(\int e^{-x+2 e^{-x} (-x+e^x (x+781250 x^2 \log (4)))} (-2+2 x+e^x (2+3125000 x \log (4))) \, dx\)

Optimal. Leaf size=21 \[ e^{2 x-2 e^{-x} x+1562500 x^2 \log (4)} \]

________________________________________________________________________________________

Rubi [F]  time = 1.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \exp \left (-x+2 e^{-x} \left (-x+e^x \left (x+781250 x^2 \log (4)\right )\right )\right ) \left (-2+2 x+e^x (2+3125000 x \log (4))\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-x + (2*(-x + E^x*(x + 781250*x^2*Log[4])))/E^x)*(-2 + 2*x + E^x*(2 + 3125000*x*Log[4])),x]

[Out]

2*Defer[Int][E^((2*x*(-1 + E^x + 781250*E^x*x*Log[4]))/E^x), x] - 2*Defer[Int][E^((x*(-2 + E^x + 1562500*E^x*x
*Log[4]))/E^x), x] + 3125000*Log[4]*Defer[Int][E^((2*x*(-1 + E^x + 781250*E^x*x*Log[4]))/E^x)*x, x] + 2*Defer[
Int][E^((x*(-2 + E^x + 1562500*E^x*x*Log[4]))/E^x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} \left (-1+e^x+x+1562500 e^x x \log (4)\right ) \, dx\\ &=2 \int e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} \left (-1+e^x+x+1562500 e^x x \log (4)\right ) \, dx\\ &=2 \int \left (-e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )}+e^{x+e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )}+e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} x+1562500 e^{x+e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} x \log (4)\right ) \, dx\\ &=-\left (2 \int e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} \, dx\right )+2 \int e^{x+e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} \, dx+2 \int e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} x \, dx+(3125000 \log (4)) \int e^{x+e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} x \, dx\\ &=2 \int e^{2 e^{-x} x \left (-1+e^x+781250 e^x x \log (4)\right )} \, dx-2 \int e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} \, dx+2 \int e^{e^{-x} x \left (-2+e^x+1562500 e^x x \log (4)\right )} x \, dx+(3125000 \log (4)) \int e^{2 e^{-x} x \left (-1+e^x+781250 e^x x \log (4)\right )} x \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [F]  time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int e^{-x+2 e^{-x} \left (-x+e^x \left (x+781250 x^2 \log (4)\right )\right )} \left (-2+2 x+e^x (2+3125000 x \log (4))\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(-x + (2*(-x + E^x*(x + 781250*x^2*Log[4])))/E^x)*(-2 + 2*x + E^x*(2 + 3125000*x*Log[4])),x]

[Out]

Integrate[E^(-x + (2*(-x + E^x*(x + 781250*x^2*Log[4])))/E^x)*(-2 + 2*x + E^x*(2 + 3125000*x*Log[4])), x]

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 24, normalized size = 1.14 \begin {gather*} e^{\left ({\left ({\left (3125000 \, x^{2} \log \relax (2) + x\right )} e^{x} - 2 \, x\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6250000*x*log(2)+2)*exp(x)+2*x-2)*exp(((1562500*x^2*log(2)+x)*exp(x)-x)/exp(x))^2/exp(x),x, algori
thm="fricas")

[Out]

e^(((3125000*x^2*log(2) + x)*e^x - 2*x)*e^(-x) + x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6250000*x*log(2)+2)*exp(x)+2*x-2)*exp(((1562500*x^2*log(2)+x)*exp(x)-x)/exp(x))^2/exp(x),x, algori
thm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 286.65, size = 20, normalized size = 0.95

method	result	size
risch	\({\mathrm e}^{2 x \left (1562500 x \ln \relax (2) {\mathrm e}^{x}+{\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}\)	\(20\)
norman	\({\mathrm e}^{2 \left (\left (1562500 x^{2} \ln \relax (2)+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6250000*x*ln(2)+2)*exp(x)+2*x-2)*exp(((1562500*x^2*ln(2)+x)*exp(x)-x)/exp(x))^2/exp(x),x,method=_RETURNV
ERBOSE)

[Out]

exp(2*x*(1562500*x*ln(2)*exp(x)+exp(x)-1)*exp(-x))

________________________________________________________________________________________

maxima [A]  time = 25.25, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (3125000 \, x^{2} \log \relax (2) - 2 \, x e^{\left (-x\right )} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6250000*x*log(2)+2)*exp(x)+2*x-2)*exp(((1562500*x^2*log(2)+x)*exp(x)-x)/exp(x))^2/exp(x),x, algori
thm="maxima")

[Out]

e^(3125000*x^2*log(2) - 2*x*e^(-x) + 2*x)

________________________________________________________________________________________

mupad [B]  time = 3.11, size = 20, normalized size = 0.95 \begin {gather*} 2^{3125000\,x^2}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(-2*exp(-x)*(x - exp(x)*(x + 1562500*x^2*log(2))))*(2*x + exp(x)*(6250000*x*log(2) + 2) - 2),x)

[Out]

2^(3125000*x^2)*exp(2*x)*exp(-2*x*exp(-x))

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 22, normalized size = 1.05 \begin {gather*} e^{2 \left (- x + \left (1562500 x^{2} \log {\relax (2 )} + x\right ) e^{x}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6250000*x*ln(2)+2)*exp(x)+2*x-2)*exp(((1562500*x**2*ln(2)+x)*exp(x)-x)/exp(x))**2/exp(x),x)

[Out]

exp(2*(-x + (1562500*x**2*log(2) + x)*exp(x))*exp(-x))

________________________________________________________________________________________