3.45.32 \(\int \frac {-81-54 x^3+5184 e^{e^x+x} x^5+108 x^6+120 x^9+32 x^{12}}{5184 x^5} \, dx\) [4432]

Optimal. Leaf size=27 \[ -5+e^{e^x}+\frac {\left (\frac {1}{2}+\frac {x^3}{3}\right )^4}{16 x^4} \]

[Out]

1/16*(1/3*x^3+1/2)^4/x^4-5+exp(exp(x))

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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {12, 14, 2320, 2225, 459} \begin {gather*} \frac {x^8}{1296}+\frac {x^5}{216}+\frac {1}{256 x^4}+\frac {x^2}{96}+e^{e^x}+\frac {1}{96 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-81 - 54*x^3 + 5184*E^(E^x + x)*x^5 + 108*x^6 + 120*x^9 + 32*x^12)/(5184*x^5),x]

[Out]

E^E^x + 1/(256*x^4) + 1/(96*x) + x^2/96 + x^5/216 + x^8/1296

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-81-54 x^3+5184 e^{e^x+x} x^5+108 x^6+120 x^9+32 x^{12}}{x^5} \, dx}{5184}\\ &=\frac {\int \left (5184 e^{e^x+x}+\frac {\left (3+2 x^3\right )^3 \left (-3+4 x^3\right )}{x^5}\right ) \, dx}{5184}\\ &=\frac {\int \frac {\left (3+2 x^3\right )^3 \left (-3+4 x^3\right )}{x^5} \, dx}{5184}+\int e^{e^x+x} \, dx\\ &=\frac {\int \left (-\frac {81}{x^5}-\frac {54}{x^2}+108 x+120 x^4+32 x^7\right ) \, dx}{5184}+\text {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}+\frac {1}{256 x^4}+\frac {1}{96 x}+\frac {x^2}{96}+\frac {x^5}{216}+\frac {x^8}{1296}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} e^{e^x}+\frac {\left (3+2 x^3\right )^4}{20736 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-81 - 54*x^3 + 5184*E^(E^x + x)*x^5 + 108*x^6 + 120*x^9 + 32*x^12)/(5184*x^5),x]

[Out]

E^E^x + (3 + 2*x^3)^4/(20736*x^4)

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Maple [A]
time = 0.21, size = 30, normalized size = 1.11

method	result	size
default	\(\frac {x^{2}}{96}+\frac {1}{256 x^{4}}+\frac {1}{96 x}+\frac {x^{5}}{216}+\frac {x^{8}}{1296}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(30\)
risch	\(\frac {x^{8}}{1296}+\frac {x^{5}}{216}+\frac {x^{2}}{96}+\frac {54 x^{3}+\frac {81}{4}}{5184 x^{4}}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5184*(5184*x^5*exp(x)*exp(exp(x))+32*x^12+120*x^9+108*x^6-54*x^3-81)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/96*x^2+1/256/x^4+1/96/x+1/216*x^5+1/1296*x^8+exp(exp(x))

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Maxima [A]
time = 0.28, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{1296} \, x^{8} + \frac {1}{216} \, x^{5} + \frac {1}{96} \, x^{2} + \frac {1}{96 \, x} + \frac {1}{256 \, x^{4}} + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5184*(5184*x^5*exp(x)*exp(exp(x))+32*x^12+120*x^9+108*x^6-54*x^3-81)/x^5,x, algorithm="maxima")

[Out]

1/1296*x^8 + 1/216*x^5 + 1/96*x^2 + 1/96/x + 1/256/x^4 + e^(e^x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (19) = 38\).
time = 0.38, size = 45, normalized size = 1.67 \begin {gather*} \frac {{\left (20736 \, x^{4} e^{\left (x + e^{x}\right )} + {\left (16 \, x^{12} + 96 \, x^{9} + 216 \, x^{6} + 216 \, x^{3} + 81\right )} e^{x}\right )} e^{\left (-x\right )}}{20736 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5184*(5184*x^5*exp(x)*exp(exp(x))+32*x^12+120*x^9+108*x^6-54*x^3-81)/x^5,x, algorithm="fricas")

[Out]

1/20736*(20736*x^4*e^(x + e^x) + (16*x^12 + 96*x^9 + 216*x^6 + 216*x^3 + 81)*e^x)*e^(-x)/x^4

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Sympy [A]
time = 0.07, size = 31, normalized size = 1.15 \begin {gather*} \frac {x^{8}}{1296} + \frac {x^{5}}{216} + \frac {x^{2}}{96} + e^{e^{x}} + \frac {216 x^{3} + 81}{20736 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5184*(5184*x**5*exp(x)*exp(exp(x))+32*x**12+120*x**9+108*x**6-54*x**3-81)/x**5,x)

[Out]

x**8/1296 + x**5/216 + x**2/96 + exp(exp(x)) + (216*x**3 + 81)/(20736*x**4)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (19) = 38\).
time = 0.42, size = 52, normalized size = 1.93 \begin {gather*} \frac {{\left (16 \, x^{12} e^{x} + 96 \, x^{9} e^{x} + 216 \, x^{6} e^{x} + 20736 \, x^{4} e^{\left (x + e^{x}\right )} + 216 \, x^{3} e^{x} + 81 \, e^{x}\right )} e^{\left (-x\right )}}{20736 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5184*(5184*x^5*exp(x)*exp(exp(x))+32*x^12+120*x^9+108*x^6-54*x^3-81)/x^5,x, algorithm="giac")

[Out]

1/20736*(16*x^12*e^x + 96*x^9*e^x + 216*x^6*e^x + 20736*x^4*e^(x + e^x) + 216*x^3*e^x + 81*e^x)*e^(-x)/x^4

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Mupad [B]
time = 3.24, size = 30, normalized size = 1.11 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}+\frac {\frac {x^3}{96}+\frac {1}{256}}{x^4}+\frac {x^2}{96}+\frac {x^5}{216}+\frac {x^8}{1296} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6/48 - x^3/96 + (5*x^9)/216 + x^12/162 + x^5*exp(exp(x))*exp(x) - 1/64)/x^5,x)

[Out]

exp(exp(x)) + (x^3/96 + 1/256)/x^4 + x^2/96 + x^5/216 + x^8/1296

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