Integrand size = 166, antiderivative size = 25 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{\left (4 e \left (1-e^x\right )+4 \left (-\frac {15}{2}+e^4\right ) x\right )^2} \]
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{4 \left (2 e-2 e^{1+x}-15 x+2 e^4 x\right )^2} \]
Integrate[(-45 + 6*E^4 - 6*E^(1 + x))/(-16*E^3 + 16*E^(3 + 3*x) + 360*E^2* x - 2700*E*x^2 + 6750*x^3 - 16*E^12*x^3 + E^(2*x)*(-48*E^3 + 360*E^2*x - 4 8*E^6*x) + E^4*(-48*E^2*x + 720*E*x^2 - 2700*x^3) + E^8*(-48*E*x^2 + 360*x ^3) + E^x*(48*E^3 - 720*E^2*x + 2700*E*x^2 + 48*E^9*x^2 + E^4*(96*E^2*x - 720*E*x^2))),x]
Time = 0.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 7239, 27, 7237}
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.
\(\displaystyle \int \frac {-6 e^{x+1}-45+6 e^4}{-16 e^{12} x^3+6750 x^3-2700 e x^2+e^x \left (48 e^9 x^2+2700 e x^2+e^4 \left (96 e^2 x-720 e x^2\right )-720 e^2 x+48 e^3\right )+e^4 \left (-2700 x^3+720 e x^2-48 e^2 x\right )+e^8 \left (360 x^3-48 e x^2\right )+360 e^2 x+16 e^{3 x+3}+e^{2 x} \left (-48 e^6 x+360 e^2 x-48 e^3\right )-16 e^3} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-6 e^{x+1}-45+6 e^4}{\left (6750-16 e^{12}\right ) x^3-2700 e x^2+e^x \left (48 e^9 x^2+2700 e x^2+e^4 \left (96 e^2 x-720 e x^2\right )-720 e^2 x+48 e^3\right )+e^4 \left (-2700 x^3+720 e x^2-48 e^2 x\right )+e^8 \left (360 x^3-48 e x^2\right )+360 e^2 x+16 e^{3 x+3}+e^{2 x} \left (-48 e^6 x+360 e^2 x-48 e^3\right )-16 e^3}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 \left (2 e^{x+1}+15 \left (1-\frac {2 e^4}{15}\right )\right )}{2 \left (-15 \left (1-\frac {2 e^4}{15}\right ) x-2 e^{x+1}+2 e\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{2} \int \frac {15-2 e^4+2 e^{x+1}}{\left (-\left (\left (15-2 e^4\right ) x\right )-2 e^{x+1}+2 e\right )^3}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {3}{4 \left (-\left (\left (15-2 e^4\right ) x\right )-2 e^{x+1}+2 e\right )^2}\) |
Int[(-45 + 6*E^4 - 6*E^(1 + x))/(-16*E^3 + 16*E^(3 + 3*x) + 360*E^2*x - 27 00*E*x^2 + 6750*x^3 - 16*E^12*x^3 + E^(2*x)*(-48*E^3 + 360*E^2*x - 48*E^6* x) + E^4*(-48*E^2*x + 720*E*x^2 - 2700*x^3) + E^8*(-48*E*x^2 + 360*x^3) + E^x*(48*E^3 - 720*E^2*x + 2700*E*x^2 + 48*E^9*x^2 + E^4*(96*E^2*x - 720*E* x^2))),x]
3.1.84.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {3}{4 \left (2 \,{\mathrm e} \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}+15 x \right )^{2}}\) | \(24\) |
risch | \(\frac {3}{4 \left (2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{1+x}+2 \,{\mathrm e}-15 x \right )^{2}}\) | \(24\) |
parallelrisch | \(\frac {3}{4 \left (4 \,{\mathrm e}^{2 x} {\mathrm e}^{2}-8 \,{\mathrm e} \,{\mathrm e}^{4} {\mathrm e}^{x} x +4 x^{2} {\mathrm e}^{8}-8 \,{\mathrm e}^{2} {\mathrm e}^{x}+8 \,{\mathrm e} \,{\mathrm e}^{4} x +60 x \,{\mathrm e} \,{\mathrm e}^{x}-60 x^{2} {\mathrm e}^{4}+4 \,{\mathrm e}^{2}-60 x \,{\mathrm e}+225 x^{2}\right )}\) | \(79\) |
int((-6*exp(1)*exp(x)+6*exp(4)-45)/(16*exp(1)^3*exp(x)^3+(-48*x*exp(1)^2*e xp(4)-48*exp(1)^3+360*x*exp(1)^2)*exp(x)^2+(48*x^2*exp(1)*exp(4)^2+(96*x*e xp(1)^2-720*x^2*exp(1))*exp(4)+48*exp(1)^3-720*x*exp(1)^2+2700*x^2*exp(1)) *exp(x)-16*x^3*exp(4)^3+(-48*x^2*exp(1)+360*x^3)*exp(4)^2+(-48*x*exp(1)^2+ 720*x^2*exp(1)-2700*x^3)*exp(4)-16*exp(1)^3+360*x*exp(1)^2-2700*x^2*exp(1) +6750*x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{4 \, {\left (4 \, x^{2} e^{8} - 60 \, x^{2} e^{4} + 225 \, x^{2} + 8 \, x e^{5} - 60 \, x e - 4 \, {\left (2 \, x e^{4} - 15 \, x + 2 \, e\right )} e^{\left (x + 1\right )} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )}\right )}} \]
integrate((-6*exp(1)*exp(x)+6*exp(4)-45)/(16*exp(1)^3*exp(x)^3+(-48*x*exp( 1)^2*exp(4)-48*exp(1)^3+360*x*exp(1)^2)*exp(x)^2+(48*x^2*exp(1)*exp(4)^2+( 96*x*exp(1)^2-720*x^2*exp(1))*exp(4)+48*exp(1)^3-720*x*exp(1)^2+2700*x^2*e xp(1))*exp(x)-16*x^3*exp(4)^3+(-48*x^2*exp(1)+360*x^3)*exp(4)^2+(-48*x*exp (1)^2+720*x^2*exp(1)-2700*x^3)*exp(4)-16*exp(1)^3+360*x*exp(1)^2-2700*x^2* exp(1)+6750*x^3),x, algorithm=\
3/4/(4*x^2*e^8 - 60*x^2*e^4 + 225*x^2 + 8*x*e^5 - 60*x*e - 4*(2*x*e^4 - 15 *x + 2*e)*e^(x + 1) + 4*e^2 + 4*e^(2*x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.92 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{- 240 x^{2} e^{4} + 900 x^{2} + 16 x^{2} e^{8} - 240 e x + 32 x e^{5} + \left (- 32 x e^{5} + 240 e x - 32 e^{2}\right ) e^{x} + 16 e^{2} e^{2 x} + 16 e^{2}} \]
integrate((-6*exp(1)*exp(x)+6*exp(4)-45)/(16*exp(1)**3*exp(x)**3+(-48*x*ex p(1)**2*exp(4)-48*exp(1)**3+360*x*exp(1)**2)*exp(x)**2+(48*x**2*exp(1)*exp (4)**2+(96*x*exp(1)**2-720*x**2*exp(1))*exp(4)+48*exp(1)**3-720*x*exp(1)** 2+2700*x**2*exp(1))*exp(x)-16*x**3*exp(4)**3+(-48*x**2*exp(1)+360*x**3)*ex p(4)**2+(-48*x*exp(1)**2+720*x**2*exp(1)-2700*x**3)*exp(4)-16*exp(1)**3+36 0*x*exp(1)**2-2700*x**2*exp(1)+6750*x**3),x)
3/(-240*x**2*exp(4) + 900*x**2 + 16*x**2*exp(8) - 240*E*x + 32*x*exp(5) + (-32*x*exp(5) + 240*E*x - 32*exp(2))*exp(x) + 16*exp(2)*exp(2*x) + 16*exp( 2))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{4 \, {\left (x^{2} {\left (4 \, e^{8} - 60 \, e^{4} + 225\right )} + 4 \, x {\left (2 \, e^{5} - 15 \, e\right )} - 4 \, {\left (x {\left (2 \, e^{5} - 15 \, e\right )} + 2 \, e^{2}\right )} e^{x} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )}\right )}} \]
integrate((-6*exp(1)*exp(x)+6*exp(4)-45)/(16*exp(1)^3*exp(x)^3+(-48*x*exp( 1)^2*exp(4)-48*exp(1)^3+360*x*exp(1)^2)*exp(x)^2+(48*x^2*exp(1)*exp(4)^2+( 96*x*exp(1)^2-720*x^2*exp(1))*exp(4)+48*exp(1)^3-720*x*exp(1)^2+2700*x^2*e xp(1))*exp(x)-16*x^3*exp(4)^3+(-48*x^2*exp(1)+360*x^3)*exp(4)^2+(-48*x*exp (1)^2+720*x^2*exp(1)-2700*x^3)*exp(4)-16*exp(1)^3+360*x*exp(1)^2-2700*x^2* exp(1)+6750*x^3),x, algorithm=\
3/4/(x^2*(4*e^8 - 60*e^4 + 225) + 4*x*(2*e^5 - 15*e) - 4*(x*(2*e^5 - 15*e) + 2*e^2)*e^x + 4*e^2 + 4*e^(2*x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=\frac {3}{4 \, {\left (4 \, x^{2} e^{8} - 60 \, x^{2} e^{4} + 225 \, x^{2} + 8 \, x e^{5} - 60 \, x e - 8 \, x e^{\left (x + 5\right )} + 60 \, x e^{\left (x + 1\right )} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )} - 8 \, e^{\left (x + 2\right )}\right )}} \]
integrate((-6*exp(1)*exp(x)+6*exp(4)-45)/(16*exp(1)^3*exp(x)^3+(-48*x*exp( 1)^2*exp(4)-48*exp(1)^3+360*x*exp(1)^2)*exp(x)^2+(48*x^2*exp(1)*exp(4)^2+( 96*x*exp(1)^2-720*x^2*exp(1))*exp(4)+48*exp(1)^3-720*x*exp(1)^2+2700*x^2*e xp(1))*exp(x)-16*x^3*exp(4)^3+(-48*x^2*exp(1)+360*x^3)*exp(4)^2+(-48*x*exp (1)^2+720*x^2*exp(1)-2700*x^3)*exp(4)-16*exp(1)^3+360*x*exp(1)^2-2700*x^2* exp(1)+6750*x^3),x, algorithm=\
3/4/(4*x^2*e^8 - 60*x^2*e^4 + 225*x^2 + 8*x*e^5 - 60*x*e - 8*x*e^(x + 5) + 60*x*e^(x + 1) + 4*e^2 + 4*e^(2*x + 2) - 8*e^(x + 2))
Time = 7.73 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.72 \[ \int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx=-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{2}-3\,{\mathrm {e}}^x-\frac {x\,{\mathrm {e}}^{x-1}\,\left (6\,{\mathrm {e}}^4-45\right )}{2}+\frac {x\,{\mathrm {e}}^{-1}\,\left (6\,{\mathrm {e}}^4-45\right )}{2}+\frac {x^2\,{\mathrm {e}}^{-2}\,\left (12\,{\mathrm {e}}^8-180\,{\mathrm {e}}^4+675\right )}{8}}{8\,{\mathrm {e}}^2-16\,{\mathrm {e}}^{x+2}+8\,{\mathrm {e}}^{2\,x+2}+120\,x\,{\mathrm {e}}^{x+1}-16\,x\,{\mathrm {e}}^{x+5}-120\,x\,\mathrm {e}+16\,x\,{\mathrm {e}}^5-120\,x^2\,{\mathrm {e}}^4+8\,x^2\,{\mathrm {e}}^8+450\,x^2} \]
int((6*exp(1)*exp(x) - 6*exp(4) + 45)/(16*exp(3) - 16*exp(3*x)*exp(3) + ex p(4)*(48*x*exp(2) - 720*x^2*exp(1) + 2700*x^3) + exp(2*x)*(48*exp(3) - 360 *x*exp(2) + 48*x*exp(6)) - 360*x*exp(2) + 2700*x^2*exp(1) + 16*x^3*exp(12) - exp(x)*(48*exp(3) - 720*x*exp(2) + 2700*x^2*exp(1) + 48*x^2*exp(9) + ex p(4)*(96*x*exp(2) - 720*x^2*exp(1))) - 6750*x^3 + exp(8)*(48*x^2*exp(1) - 360*x^3)),x)
-((3*exp(2*x))/2 - 3*exp(x) - (x*exp(x - 1)*(6*exp(4) - 45))/2 + (x*exp(-1 )*(6*exp(4) - 45))/2 + (x^2*exp(-2)*(12*exp(8) - 180*exp(4) + 675))/8)/(8* exp(2) - 16*exp(x + 2) + 8*exp(2*x + 2) + 120*x*exp(x + 1) - 16*x*exp(x + 5) - 120*x*exp(1) + 16*x*exp(5) - 120*x^2*exp(4) + 8*x^2*exp(8) + 450*x^2)