Integrand size = 127, antiderivative size = 38 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=e^{\frac {e^{5+x} \left (4+\frac {1}{9} (5-x)^2\right )}{x+x \left (\frac {e^{2 x}}{4}+x\right )}} \]
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=e^{\frac {4 e^{5+x} \left (61-10 x+x^2\right )}{9 x \left (4+e^{2 x}+4 x\right )}} \]
Integrate[(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))*(-976 - 976*x + 992*x^2 - 144*x^3 + 16*x^4 + E^(2*x)*(-244 - 2 44*x + 44*x^2 - 4*x^3)))/(144*x^2 + 9*E^(4*x)*x^2 + 288*x^3 + 144*x^4 + E^ (2*x)*(72*x^2 + 72*x^3)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (16 x^4-144 x^3+992 x^2+e^{2 x} \left (-4 x^3+44 x^2-244 x-244\right )-976 x-976\right ) \exp \left (\frac {e^{x+5} \left (4 x^2-40 x+244\right )}{36 x^2+9 e^{2 x} x+36 x}+x+5\right )}{144 x^4+288 x^3+9 e^{4 x} x^2+144 x^2+e^{2 x} \left (72 x^3+72 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (16 x^4-144 x^3+992 x^2+e^{2 x} \left (-4 x^3+44 x^2-244 x-244\right )-976 x-976\right ) \exp \left (\frac {e^{x+5} \left (4 x^2-40 x+244\right )}{36 x^2+9 e^{2 x} x+36 x}+x+5\right )}{9 x^2 \left (4 x+e^{2 x}+4\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {4 \exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) \left (-4 x^4+36 x^3-248 x^2+244 x+e^{2 x} \left (x^3-11 x^2+61 x+61\right )+244\right )}{x^2 \left (4 x+e^{2 x}+4\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4}{9} \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) \left (-4 x^4+36 x^3-248 x^2+244 x+e^{2 x} \left (x^3-11 x^2+61 x+61\right )+244\right )}{x^2 \left (4 x+e^{2 x}+4\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4}{9} \int \left (\frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) \left (x^3-11 x^2+61 x+61\right )}{x^2 \left (4 x+e^{2 x}+4\right )}-\frac {4 \exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) \left (2 x^3-19 x^2+112 x+61\right )}{x \left (4 x+e^{2 x}+4\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{9} \left (-448 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right )}{\left (4 x+e^{2 x}+4\right )^2}dx-244 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right )}{x \left (4 x+e^{2 x}+4\right )^2}dx+76 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) x}{\left (4 x+e^{2 x}+4\right )^2}dx-8 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) x^2}{\left (4 x+e^{2 x}+4\right )^2}dx-11 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right )}{4 x+e^{2 x}+4}dx+61 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right )}{x^2 \left (4 x+e^{2 x}+4\right )}dx+61 \int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right )}{x \left (4 x+e^{2 x}+4\right )}dx+\int \frac {\exp \left (x+\frac {4 e^{x+5} \left (x^2-10 x+61\right )}{9 \left (4 x^2+e^{2 x} x+4 x\right )}+5\right ) x}{4 x+e^{2 x}+4}dx\right )\) |
Int[(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36* x^2))*(-976 - 976*x + 992*x^2 - 144*x^3 + 16*x^4 + E^(2*x)*(-244 - 244*x + 44*x^2 - 4*x^3)))/(144*x^2 + 9*E^(4*x)*x^2 + 288*x^3 + 144*x^4 + E^(2*x)* (72*x^2 + 72*x^3)),x]
3.2.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 75.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79
method | result | size |
risch | \({\mathrm e}^{\frac {4 \left (x^{2}-10 x +61\right ) {\mathrm e}^{5+x}}{9 x \left ({\mathrm e}^{2 x}+4 x +4\right )}}\) | \(30\) |
parallelrisch | \({\mathrm e}^{\frac {\left (4 x^{2}-40 x +244\right ) {\mathrm e}^{5+x}}{9 x \left ({\mathrm e}^{2 x}+4 x +4\right )}}\) | \(32\) |
int(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x-976)* exp(5+x)*exp((4*x^2-40*x+244)*exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/(9*x^2* exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x,method=_RETUR NVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.95 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=e^{\left (-x + \frac {36 \, {\left (x^{3} + 6 \, x^{2} + 5 \, x\right )} e^{10} + 9 \, {\left (x^{2} + 5 \, x\right )} e^{\left (2 \, x + 10\right )} + 4 \, {\left (x^{2} - 10 \, x + 61\right )} e^{\left (x + 15\right )}}{9 \, {\left (4 \, {\left (x^{2} + x\right )} e^{10} + x e^{\left (2 \, x + 10\right )}\right )}} - 5\right )} \]
integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x -976)*exp(5+x)*exp((4*x^2-40*x+244)*exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/( 9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algori thm=\
e^(-x + 1/9*(36*(x^3 + 6*x^2 + 5*x)*e^10 + 9*(x^2 + 5*x)*e^(2*x + 10) + 4* (x^2 - 10*x + 61)*e^(x + 15))/(4*(x^2 + x)*e^10 + x*e^(2*x + 10)) - 5)
Time = 0.44 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=e^{\frac {\left (4 x^{2} - 40 x + 244\right ) e^{5} \sqrt {e^{2 x}}}{36 x^{2} + 9 x e^{2 x} + 36 x}} \]
integrate(((-4*x**3+44*x**2-244*x-244)*exp(x)**2+16*x**4-144*x**3+992*x**2 -976*x-976)*exp(5+x)*exp((4*x**2-40*x+244)*exp(5+x)/(9*x*exp(x)**2+36*x**2 +36*x))/(9*x**2*exp(x)**4+(72*x**3+72*x**2)*exp(x)**2+144*x**4+288*x**3+14 4*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (30) = 60\).
Time = 0.66 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=e^{\left (-\frac {e^{\left (3 \, x + 5\right )}}{9 \, {\left (4 \, x + e^{\left (2 \, x\right )} + 4\right )}} - \frac {976 \, e^{\left (x + 5\right )}}{9 \, {\left (4 \, {\left (x + 2\right )} e^{\left (2 \, x\right )} + 16 \, x + e^{\left (4 \, x\right )} + 16\right )}} + \frac {244 \, e^{\left (x + 5\right )}}{9 \, {\left (x e^{\left (2 \, x\right )} + 4 \, x\right )}} - \frac {44 \, e^{\left (x + 5\right )}}{9 \, {\left (4 \, x + e^{\left (2 \, x\right )} + 4\right )}} + \frac {1}{9} \, e^{\left (x + 5\right )}\right )} \]
integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x -976)*exp(5+x)*exp((4*x^2-40*x+244)*exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/( 9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algori thm=\
e^(-1/9*e^(3*x + 5)/(4*x + e^(2*x) + 4) - 976/9*e^(x + 5)/(4*(x + 2)*e^(2* x) + 16*x + e^(4*x) + 16) + 244/9*e^(x + 5)/(x*e^(2*x) + 4*x) - 44/9*e^(x + 5)/(4*x + e^(2*x) + 4) + 1/9*e^(x + 5))
Exception generated. \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x -976)*exp(5+x)*exp((4*x^2-40*x+244)*exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/( 9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algori thm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{3962711310336,[0,19,36,2]%%%}+%%%{-974826982342656,[0,19,3 5,2]%%%}+
Time = 10.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {e^{5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}} \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx={\mathrm {e}}^{-\frac {40\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,{\mathrm {e}}^{2\,x}+36}}\,{\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,{\mathrm {e}}^{2\,x}+36}}\,{\mathrm {e}}^{\frac {244\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,x\,{\mathrm {e}}^{2\,x}+36\,x^2}} \]
int(-(exp(x + 5)*exp((exp(x + 5)*(4*x^2 - 40*x + 244))/(36*x + 9*x*exp(2*x ) + 36*x^2))*(976*x + exp(2*x)*(244*x - 44*x^2 + 4*x^3 + 244) - 992*x^2 + 144*x^3 - 16*x^4 + 976))/(exp(2*x)*(72*x^2 + 72*x^3) + 9*x^2*exp(4*x) + 14 4*x^2 + 288*x^3 + 144*x^4),x)