3.12.74 \(\int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 (576+5670 x-15552 x^2)+e^2 (1152 x+11016 x^2-10368 x^3)}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 (-1152 x^3-11664 x^4+31104 x^5)+e^2 (-2304 x^4-23328 x^5+20736 x^6)} \, dx\) [1174]

Optimal. Leaf size=27 \[ \frac {2}{x^2 \left (8+\frac {x}{\frac {1}{9}+x-\left (e^2+x\right )^2}\right )} \]

[Out]

2/(x/(x-(x+exp(2))^2+1/9)+8)/x^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(27)=54\).
time = 0.26, antiderivative size = 95, normalized size of antiderivative = 3.52, number of steps used = 10, number of rules used = 5, integrand size = 148, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 2099, 652, 632, 210} \begin {gather*} \frac {81 \left (-8 \left (985-2592 e^2\right ) x+41472 e^4-39088 e^2+8865\right )}{32 \left (985-2592 e^2\right ) \left (1-9 e^4\right ) \left (-72 x^2+9 \left (9-16 e^2\right ) x+8 \left (1-9 e^4\right )\right )}+\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32 - 2592*E^8 - 630*x - 10368*E^6*x - 2340*x^2 + 5346*x^3 - 2592*x^4 + E^4*(576 + 5670*x - 15552*x^2) +
E^2*(1152*x + 11016*x^2 - 10368*x^3))/(64*x^3 + 5184*E^8*x^3 + 1296*x^4 + 20736*E^6*x^4 + 5409*x^5 - 11664*x^6
+ 5184*x^7 + E^4*(-1152*x^3 - 11664*x^4 + 31104*x^5) + E^2*(-2304*x^4 - 23328*x^5 + 20736*x^6)),x]

[Out]

1/(4*x^2) - 9/(32*(1 - 9*E^4)*x) + (81*(8865 - 39088*E^2 + 41472*E^4 - 8*(985 - 2592*E^2)*x))/(32*(985 - 2592*
E^2)*(1 - 9*E^4)*(8*(1 - 9*E^4) + 9*(9 - 16*E^2)*x - 72*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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32-2592 e^8+\left (-630-10368 e^6\right ) x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx\\ &=\int \frac {-32-2592 e^8+\left (-630-10368 e^6\right ) x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{\left (64+5184 e^8\right ) x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx\\ &=\int \frac {-32-2592 e^8+\left (-630-10368 e^6\right ) x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{\left (64+5184 e^8\right ) x^3+\left (1296+20736 e^6\right ) x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx\\ &=\int \left (-\frac {1}{2 x^3}-\frac {9}{32 \left (-1+9 e^4\right ) x^2}+\frac {81 \left (-857+2592 e^2-1152 e^4+72 \left (9-16 e^2\right ) x\right )}{32 \left (1-9 e^4\right ) \left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )^2}+\frac {81}{4 \left (1-3 e^2\right ) \left (1+3 e^2\right ) \left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )}\right ) \, dx\\ &=\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x}+\frac {81 \int \frac {-857+2592 e^2-1152 e^4+72 \left (9-16 e^2\right ) x}{\left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )^2} \, dx}{32 \left (1-9 e^4\right )}+\frac {81 \int \frac {1}{8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2} \, dx}{4 \left (1-9 e^4\right )}\\ &=\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x}+\frac {81 \left (8865-39088 e^2+41472 e^4-8 \left (985-2592 e^2\right ) x\right )}{32 \left (985-2592 e^2\right ) \left (1-9 e^4\right ) \left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )}-\frac {81 \int \frac {1}{8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2} \, dx}{4 \left (1-9 e^4\right )}-\frac {81 \text {Subst}\left (\int \frac {1}{9 \left (985-2592 e^2\right )-x^2} \, dx,x,9 \left (9-16 e^2\right )-144 x\right )}{2 \left (1-9 e^4\right )}\\ &=\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x}+\frac {81 \left (8865-39088 e^2+41472 e^4-8 \left (985-2592 e^2\right ) x\right )}{32 \left (985-2592 e^2\right ) \left (1-9 e^4\right ) \left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )}+\frac {27 \tan ^{-1}\left (\frac {3 \left (9-16 e^2-16 x\right )}{\sqrt {-985+2592 e^2}}\right )}{2 \sqrt {-985+2592 e^2} \left (1-9 e^4\right )}+\frac {81 \text {Subst}\left (\int \frac {1}{9 \left (985-2592 e^2\right )-x^2} \, dx,x,9 \left (9-16 e^2\right )-144 x\right )}{2 \left (1-9 e^4\right )}\\ &=\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x}+\frac {81 \left (8865-39088 e^2+41472 e^4-8 \left (985-2592 e^2\right ) x\right )}{32 \left (985-2592 e^2\right ) \left (1-9 e^4\right ) \left (8 \left (1-9 e^4\right )+9 \left (9-16 e^2\right ) x-72 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 49, normalized size = 1.81 \begin {gather*} -\frac {2 \left (1-9 e^4+9 x-18 e^2 x-9 x^2\right )}{x^2 \left (-8+72 e^4-81 x+144 e^2 x+72 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32 - 2592*E^8 - 630*x - 10368*E^6*x - 2340*x^2 + 5346*x^3 - 2592*x^4 + E^4*(576 + 5670*x - 15552*x
^2) + E^2*(1152*x + 11016*x^2 - 10368*x^3))/(64*x^3 + 5184*E^8*x^3 + 1296*x^4 + 20736*E^6*x^4 + 5409*x^5 - 116
64*x^6 + 5184*x^7 + E^4*(-1152*x^3 - 11664*x^4 + 31104*x^5) + E^2*(-2304*x^4 - 23328*x^5 + 20736*x^6)),x]

[Out]

(-2*(1 - 9*E^4 + 9*x - 18*E^2*x - 9*x^2))/(x^2*(-8 + 72*E^4 - 81*x + 144*E^2*x + 72*x^2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 280, normalized size = 10.37

method	result	size
risch	\(\frac {-2+18 x^{2}+72 \left (\frac {{\mathrm e}^{2}}{2}-\frac {1}{4}\right ) x +18 \,{\mathrm e}^{4}}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\)	\(46\)
norman	\(\frac {\left (36 \,{\mathrm e}^{2}-18\right ) x +18 x^{2}-2+18 \,{\mathrm e}^{4}}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\)	\(49\)
gosper	\(\frac {18 \,{\mathrm e}^{4}+36 \,{\mathrm e}^{2} x +18 x^{2}-18 x -2}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\)	\(50\)
default	\(\frac {9 \left (\munderset {\textit {\_R} =\RootOf \left (5184 \textit {\_Z}^{4}+\left (20736 \,{\mathrm e}^{2}-11664\right ) \textit {\_Z}^{3}+\left (-23328 \,{\mathrm e}^{2}+31104 \,{\mathrm e}^{4}+5409\right ) \textit {\_Z}^{2}+\left (-2304 \,{\mathrm e}^{2}-11664 \,{\mathrm e}^{4}+20736 \,{\mathrm e}^{6}+1296\right ) \textit {\_Z} -1152 \,{\mathrm e}^{4}+5184 \,{\mathrm e}^{8}+64\right )}{\sum }\frac {\left (-793+576 \left (-1+27 \,{\mathrm e}^{4}+729 \,{\mathrm e}^{12}-243 \,{\mathrm e}^{8}\right ) \textit {\_R}^{2}+144 \left (9-16 \,{\mathrm e}^{2}-243 \,{\mathrm e}^{4}-6561 \,{\mathrm e}^{12}-3888 \,{\mathrm e}^{10}+11664 \,{\mathrm e}^{14}+2187 \,{\mathrm e}^{8}+432 \,{\mathrm e}^{6}\right ) \textit {\_R} +2592 \,{\mathrm e}^{2}+19683 \,{\mathrm e}^{4}+158193 \,{\mathrm e}^{12}+629856 \,{\mathrm e}^{10}-1889568 \,{\mathrm e}^{14}+1259712 \,{\mathrm e}^{16}-146043 \,{\mathrm e}^{8}-69984 \,{\mathrm e}^{6}\right ) \ln \left (x -\textit {\_R} \right )}{72+1152 \,{\mathrm e}^{6}+3456 \textit {\_R} \,{\mathrm e}^{4}+3456 \textit {\_R}^{2} {\mathrm e}^{2}+1152 \textit {\_R}^{3}-648 \,{\mathrm e}^{4}-2592 \,{\mathrm e}^{2} \textit {\_R} -1944 \textit {\_R}^{2}-128 \,{\mathrm e}^{2}+601 \textit {\_R}}\right )}{64 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2}}-\frac {36 \,{\mathrm e}^{4}+2916 \,{\mathrm e}^{12}-6561 \,{\mathrm e}^{16}-486 \,{\mathrm e}^{8}-1}{4 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2} x^{2}}-\frac {-243 \,{\mathrm e}^{4}-6561 \,{\mathrm e}^{12}+2187 \,{\mathrm e}^{8}+9}{32 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2} x}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^2+(-10368*x^3+11016*x^2+1152*x)*exp(2)-259
2*x^4+5346*x^3-2340*x^2-630*x-32)/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*exp(2)^
2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+1296*x^4+64*x^3),x,method=_RETURNVERBOSE)

[Out]

9/64/(18*exp(4)-81*exp(8)-1)^2*sum((-793+576*(-1+27*exp(4)+729*exp(12)-243*exp(8))*_R^2+144*(9-16*exp(2)-243*e
xp(4)-6561*exp(12)-3888*exp(10)+11664*exp(14)+2187*exp(8)+432*exp(6))*_R+2592*exp(2)+19683*exp(4)+158193*exp(1
2)+629856*exp(10)-1889568*exp(14)+1259712*exp(16)-146043*exp(8)-69984*exp(6))/(72+1152*exp(6)+3456*_R*exp(4)+3
456*_R^2*exp(2)+1152*_R^3-648*exp(4)-2592*exp(2)*_R-1944*_R^2-128*exp(2)+601*_R)*ln(x-_R),_R=RootOf(5184*_Z^4+
(20736*exp(2)-11664)*_Z^3+(-23328*exp(2)+31104*exp(4)+5409)*_Z^2+(-2304*exp(2)-11664*exp(4)+20736*exp(6)+1296)
*_Z-1152*exp(4)+5184*exp(8)+64))-1/4*(36*exp(4)+2916*exp(12)-6561*exp(16)-486*exp(8)-1)/(18*exp(4)-81*exp(8)-1
)^2/x^2-1/32/(18*exp(4)-81*exp(8)-1)^2*(-243*exp(4)-6561*exp(12)+2187*exp(8)+9)/x

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Maxima [A]
time = 0.27, size = 52, normalized size = 1.93 \begin {gather*} \frac {2 \, {\left (9 \, x^{2} + 9 \, x {\left (2 \, e^{2} - 1\right )} + 9 \, e^{4} - 1\right )}}{72 \, x^{4} + 9 \, x^{3} {\left (16 \, e^{2} - 9\right )} + 8 \, x^{2} {\left (9 \, e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^2+(-10368*x^3+11016*x^2+1152*x)*exp(
2)-2592*x^4+5346*x^3-2340*x^2-630*x-32)/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*e
xp(2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+1296*x^4+64*x^3),x, algorithm="maxim
a")

[Out]

2*(9*x^2 + 9*x*(2*e^2 - 1) + 9*e^4 - 1)/(72*x^4 + 9*x^3*(16*e^2 - 9) + 8*x^2*(9*e^4 - 1))

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Fricas [A]
time = 0.34, size = 53, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (9 \, x^{2} + 18 \, x e^{2} - 9 \, x + 9 \, e^{4} - 1\right )}}{72 \, x^{4} + 144 \, x^{3} e^{2} - 81 \, x^{3} + 72 \, x^{2} e^{4} - 8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^2+(-10368*x^3+11016*x^2+1152*x)*exp(
2)-2592*x^4+5346*x^3-2340*x^2-630*x-32)/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*e
xp(2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+1296*x^4+64*x^3),x, algorithm="frica
s")

[Out]

2*(9*x^2 + 18*x*e^2 - 9*x + 9*e^4 - 1)/(72*x^4 + 144*x^3*e^2 - 81*x^3 + 72*x^2*e^4 - 8*x^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
time = 1.97, size = 46, normalized size = 1.70 \begin {gather*} - \frac {- 18 x^{2} + x \left (18 - 36 e^{2}\right ) - 18 e^{4} + 2}{72 x^{4} + x^{3} \left (-81 + 144 e^{2}\right ) + x^{2} \left (-8 + 72 e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2592*exp(2)**4-10368*x*exp(2)**3+(-15552*x**2+5670*x+576)*exp(2)**2+(-10368*x**3+11016*x**2+1152*x
)*exp(2)-2592*x**4+5346*x**3-2340*x**2-630*x-32)/(5184*x**3*exp(2)**4+20736*x**4*exp(2)**3+(31104*x**5-11664*x
**4-1152*x**3)*exp(2)**2+(20736*x**6-23328*x**5-2304*x**4)*exp(2)+5184*x**7-11664*x**6+5409*x**5+1296*x**4+64*
x**3),x)

[Out]

-(-18*x**2 + x*(18 - 36*exp(2)) - 18*exp(4) + 2)/(72*x**4 + x**3*(-81 + 144*exp(2)) + x**2*(-8 + 72*exp(4)))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
time = 0.41, size = 63, normalized size = 2.33 \begin {gather*} -\frac {81 \, {\left (8 \, x + 16 \, e^{2} - 9\right )}}{32 \, {\left (72 \, x^{2} + 144 \, x e^{2} - 81 \, x + 72 \, e^{4} - 8\right )} {\left (9 \, e^{4} - 1\right )}} + \frac {9 \, x + 72 \, e^{4} - 8}{32 \, x^{2} {\left (9 \, e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^2+(-10368*x^3+11016*x^2+1152*x)*exp(
2)-2592*x^4+5346*x^3-2340*x^2-630*x-32)/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*e
xp(2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+1296*x^4+64*x^3),x, algorithm="giac"
)

[Out]

-81/32*(8*x + 16*e^2 - 9)/((72*x^2 + 144*x*e^2 - 81*x + 72*e^4 - 8)*(9*e^4 - 1)) + 1/32*(9*x + 72*e^4 - 8)/(x^
2*(9*e^4 - 1))

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Mupad [B]
time = 0.98, size = 48, normalized size = 1.78 \begin {gather*} \frac {18\,x^2+\left (36\,{\mathrm {e}}^2-18\right )\,x+18\,{\mathrm {e}}^4-2}{72\,x^4+\left (144\,{\mathrm {e}}^2-81\right )\,x^3+\left (72\,{\mathrm {e}}^4-8\right )\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(630*x + 2592*exp(8) - exp(4)*(5670*x - 15552*x^2 + 576) + 10368*x*exp(6) - exp(2)*(1152*x + 11016*x^2 -
10368*x^3) + 2340*x^2 - 5346*x^3 + 2592*x^4 + 32)/(20736*x^4*exp(6) + 5184*x^3*exp(8) - exp(4)*(1152*x^3 + 116
64*x^4 - 31104*x^5) - exp(2)*(2304*x^4 + 23328*x^5 - 20736*x^6) + 64*x^3 + 1296*x^4 + 5409*x^5 - 11664*x^6 + 5
184*x^7),x)

[Out]

(18*exp(4) + 18*x^2 + x*(36*exp(2) - 18) - 2)/(x^2*(72*exp(4) - 8) + x^3*(144*exp(2) - 81) + 72*x^4)

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