3.61.84 \(\int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+6750 x^3-16 e^{12} x^3+e^{2 x} (-48 e^3+360 e^2 x-48 e^6 x)+e^4 (-48 e^2 x+720 e x^2-2700 x^3)+e^8 (-48 e x^2+360 x^3)+e^x (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 (96 e^2 x-720 e x^2))} \, dx\)

Optimal. Leaf size=25 \[ \frac {3}{\left (4 e \left (1-e^x\right )+4 \left (-\frac {15}{2}+e^4\right ) x\right )^2} \]

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Rubi [A]  time = 0.30, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 166, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6688, 12, 6686} \begin {gather*} \frac {3}{4 \left (-\left (\left (15-2 e^4\right ) x\right )-2 e^{x+1}+2 e\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-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 - 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]

[Out]

3/(4*(2*E - 2*E^(1 + x) - (15 - 2*E^4)*x)^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\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+\left (6750-16 e^{12}\right ) 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\\ &=\int \frac {3 \left (2 e^{1+x}+15 \left (1-\frac {2 e^4}{15}\right )\right )}{2 \left (2 e-2 e^{1+x}-15 \left (1-\frac {2 e^4}{15}\right ) x\right )^3} \, dx\\ &=\frac {3}{2} \int \frac {2 e^{1+x}+15 \left (1-\frac {2 e^4}{15}\right )}{\left (2 e-2 e^{1+x}-15 \left (1-\frac {2 e^4}{15}\right ) x\right )^3} \, dx\\ &=\frac {3}{4 \left (2 e-2 e^{1+x}-\left (15-2 e^4\right ) x\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.04 \begin {gather*} \frac {3}{4 \left (2 e-2 e^{1+x}+\left (-15+2 e^4\right ) x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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 - 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]

[Out]

3/(4*(2*E - 2*E^(1 + x) + (-15 + 2*E^4)*x)^2)

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fricas [B]  time = 0.65, size = 65, normalized size = 2.60 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
*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, algorithm="fricas")

[Out]

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))

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giac [B]  time = 0.31, size = 66, normalized size = 2.64 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
*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, algorithm="giac")

[Out]

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))

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maple [A]  time = 0.75, size = 24, normalized size = 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	\(\frac {3}{4 \left (2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{x +1}+2 \,{\mathrm e}-15 x \right )^{2}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-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)*ex
p(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)*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)

[Out]

3/4/(2*exp(1)*exp(x)-2*x*exp(4)-2*exp(1)+15*x)^2

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maxima [B]  time = 0.48, size = 63, normalized size = 2.52 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
*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, algorithm="maxima")

[Out]

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
))

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mupad [B]  time = 7.79, size = 118, normalized size = 4.72 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*exp(1)*exp(x) - 6*exp(4) + 45)/(16*exp(3) - 16*exp(3*x)*exp(3) + exp(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(1
2) - exp(x)*(48*exp(3) - 720*x*exp(2) + 2700*x^2*exp(1) + 48*x^2*exp(9) + exp(4)*(96*x*exp(2) - 720*x^2*exp(1)
)) - 6750*x^3 + exp(8)*(48*x^2*exp(1) - 360*x^3)),x)

[Out]

-((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)

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sympy [B]  time = 0.28, size = 73, normalized size = 2.92 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*ex
p(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)*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)

[Out]

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))

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