3.17.80 \(\int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 (-27 x^3-9 x^4)+e^4 (144+81 x^3+54 x^4+9 x^5)}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 (-9 x^3-3 x^4)+e^4 (27 x^3+18 x^4+3 x^5)} \, dx\) [1680]

Optimal. Leaf size=22 \[ 3 x-\frac {8}{\left (x+\frac {1}{3} x \left (-e^4+x\right )\right )^2} \]

[Out]

-8/(x+x*(1/3*x-1/3*exp(4)))^2+3*x

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(22)=44\).
time = 0.12, antiderivative size = 74, normalized size of antiderivative = 3.36, number of steps used = 4, number of rules used = 2, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6, 2099} \begin {gather*} -\frac {72}{\left (3-e^4\right )^2 x^2}+3 x-\frac {144}{\left (3-e^4\right )^3 \left (x-e^4+3\right )}-\frac {72}{\left (3-e^4\right )^2 \left (x-e^4+3\right )^2}+\frac {144}{\left (3-e^4\right )^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-432 - 288*x - 81*x^3 + 3*E^12*x^3 - 81*x^4 - 27*x^5 - 3*x^6 + E^8*(-27*x^3 - 9*x^4) + E^4*(144 + 81*x^3
+ 54*x^4 + 9*x^5))/(-27*x^3 + E^12*x^3 - 27*x^4 - 9*x^5 - x^6 + E^8*(-9*x^3 - 3*x^4) + E^4*(27*x^3 + 18*x^4 +
3*x^5)),x]

[Out]

-72/((3 - E^4)^2*x^2) + 144/((3 - E^4)^3*x) + 3*x - 72/((3 - E^4)^2*(3 - E^4 + x)^2) - 144/((3 - E^4)^3*(3 - E
^4 + x))

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 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 {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{\left (-27+e^{12}\right ) x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx\\ &=\int \frac {-432-288 x+\left (-81+3 e^{12}\right ) x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{\left (-27+e^{12}\right ) x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx\\ &=\int \left (3-\frac {144}{\left (-3+e^4\right )^2 \left (-3+e^4-x\right )^3}-\frac {144}{\left (-3+e^4\right )^3 \left (-3+e^4-x\right )^2}+\frac {144}{\left (-3+e^4\right )^2 x^3}+\frac {144}{\left (-3+e^4\right )^3 x^2}\right ) \, dx\\ &=-\frac {72}{\left (3-e^4\right )^2 x^2}+\frac {144}{\left (3-e^4\right )^3 x}+3 x-\frac {72}{\left (3-e^4\right )^2 \left (3-e^4+x\right )^2}-\frac {144}{\left (3-e^4\right )^3 \left (3-e^4+x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 41, normalized size = 1.86 \begin {gather*} \frac {3 \left (-24+\left (-3+e^4\right )^2 x^3-2 \left (-3+e^4\right ) x^4+x^5\right )}{x^2 \left (3-e^4+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-432 - 288*x - 81*x^3 + 3*E^12*x^3 - 81*x^4 - 27*x^5 - 3*x^6 + E^8*(-27*x^3 - 9*x^4) + E^4*(144 + 8
1*x^3 + 54*x^4 + 9*x^5))/(-27*x^3 + E^12*x^3 - 27*x^4 - 9*x^5 - x^6 + E^8*(-9*x^3 - 3*x^4) + E^4*(27*x^3 + 18*
x^4 + 3*x^5)),x]

[Out]

(3*(-24 + (-3 + E^4)^2*x^3 - 2*(-3 + E^4)*x^4 + x^5))/(x^2*(3 - E^4 + x)^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 248, normalized size = 11.27

method	result	size
risch	\(3 x -\frac {72}{x^{2} \left ({\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}-6 \,{\mathrm e}^{4}+6 x +9\right )}\)	\(31\)
norman	\(\frac {-72+\left (-9 \,{\mathrm e}^{8}+54 \,{\mathrm e}^{4}-81\right ) x^{3}+\left (6 \,{\mathrm e}^{12}-54 \,{\mathrm e}^{8}+162 \,{\mathrm e}^{4}-162\right ) x^{2}+3 x^{5}}{x^{2} \left ({\mathrm e}^{4}-x -3\right )^{2}}\)	\(59\)
gosper	\(\frac {6 x^{2} {\mathrm e}^{12}-9 \,{\mathrm e}^{8} x^{3}+3 x^{5}-54 x^{2} {\mathrm e}^{8}+54 x^{3} {\mathrm e}^{4}+162 x^{2} {\mathrm e}^{4}-81 x^{3}-162 x^{2}-72}{x^{2} \left ({\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}-6 \,{\mathrm e}^{4}+6 x +9\right )}\)	\(85\)
default	\(3 x -\frac {48 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{4}+9\right ) \textit {\_Z}^{2}+\left (-18 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{8}+27\right ) \textit {\_Z} -27 \,{\mathrm e}^{4}+9 \,{\mathrm e}^{8}-{\mathrm e}^{12}+27\right )}{\sum }\frac {\left (4374+135 \textit {\_R} \,{\mathrm e}^{16}-1458 \textit {\_R} \,{\mathrm e}^{4}+1215 \textit {\_R} \,{\mathrm e}^{8}-540 \textit {\_R} \,{\mathrm e}^{12}-18 \textit {\_R} \,{\mathrm e}^{20}+{\mathrm e}^{24} \textit {\_R} +1890 \,{\mathrm e}^{16}-10206 \,{\mathrm e}^{4}+10206 \,{\mathrm e}^{8}-5670 \,{\mathrm e}^{12}-378 \,{\mathrm e}^{20}+42 \,{\mathrm e}^{24}-2 \,{\mathrm e}^{28}+729 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{9+{\mathrm e}^{8}-2 \textit {\_R} \,{\mathrm e}^{4}+\textit {\_R}^{2}-6 \,{\mathrm e}^{4}+6 \textit {\_R}}\right )}{\left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3}}-\frac {3 \left (244944 \,{\mathrm e}^{4}-244944 \,{\mathrm e}^{8}+136080 \,{\mathrm e}^{12}-45360 \,{\mathrm e}^{16}+9072 \,{\mathrm e}^{20}-1008 \,{\mathrm e}^{24}+48 \,{\mathrm e}^{28}-104976\right )}{2 \left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3} x^{2}}-\frac {3 \left (-69984 \,{\mathrm e}^{4}+58320 \,{\mathrm e}^{8}-25920 \,{\mathrm e}^{12}+6480 \,{\mathrm e}^{16}-864 \,{\mathrm e}^{20}+48 \,{\mathrm e}^{24}+34992\right )}{\left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3} x}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+144)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3-288*x
-432)/(x^3*exp(4)^3+(-3*x^4-9*x^3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x,method=_RE
TURNVERBOSE)

[Out]

3*x-48*sum((4374+135*_R*exp(16)-1458*_R*exp(4)+1215*_R*exp(8)-540*_R*exp(12)-18*_R*exp(20)+exp(24)*_R+1890*exp
(16)-10206*exp(4)+10206*exp(8)-5670*exp(12)-378*exp(20)+42*exp(24)-2*exp(28)+729*_R)/(9+exp(8)-2*_R*exp(4)+_R^
2-6*exp(4)+6*_R)*ln(x-_R),_R=RootOf(_Z^3+(-3*exp(4)+9)*_Z^2+(-18*exp(4)+3*exp(8)+27)*_Z-27*exp(4)+9*exp(8)-exp
(12)+27))/(27*exp(4)+exp(12)-9*exp(8)-27)^3-3/2/(27*exp(4)+exp(12)-9*exp(8)-27)^3*(244944*exp(4)-244944*exp(8)
+136080*exp(12)-45360*exp(16)+9072*exp(20)-1008*exp(24)+48*exp(28)-104976)/x^2-3/(27*exp(4)+exp(12)-9*exp(8)-2
7)^3*(-69984*exp(4)+58320*exp(8)-25920*exp(12)+6480*exp(16)-864*exp(20)+48*exp(24)+34992)/x

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Maxima [A]
time = 0.28, size = 33, normalized size = 1.50 \begin {gather*} 3 \, x - \frac {72}{x^{4} - 2 \, x^{3} {\left (e^{4} - 3\right )} + x^{2} {\left (e^{8} - 6 \, e^{4} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+144)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3
-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, alg
orithm="maxima")

[Out]

3*x - 72/(x^4 - 2*x^3*(e^4 - 3) + x^2*(e^8 - 6*e^4 + 9))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
time = 0.33, size = 71, normalized size = 3.23 \begin {gather*} \frac {3 \, {\left (x^{5} + 6 \, x^{4} + x^{3} e^{8} + 9 \, x^{3} - 2 \, {\left (x^{4} + 3 \, x^{3}\right )} e^{4} - 24\right )}}{x^{4} + 6 \, x^{3} + x^{2} e^{8} + 9 \, x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+144)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3
-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, alg
orithm="fricas")

[Out]

3*(x^5 + 6*x^4 + x^3*e^8 + 9*x^3 - 2*(x^4 + 3*x^3)*e^4 - 24)/(x^4 + 6*x^3 + x^2*e^8 + 9*x^2 - 2*(x^3 + 3*x^2)*
e^4)

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Sympy [A]
time = 0.36, size = 31, normalized size = 1.41 \begin {gather*} 3 x - \frac {72}{x^{4} + x^{3} \cdot \left (6 - 2 e^{4}\right ) + x^{2} \left (- 6 e^{4} + 9 + e^{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3*exp(4)**3+(-9*x**4-27*x**3)*exp(4)**2+(9*x**5+54*x**4+81*x**3+144)*exp(4)-3*x**6-27*x**5-81*
x**4-81*x**3-288*x-432)/(x**3*exp(4)**3+(-3*x**4-9*x**3)*exp(4)**2+(3*x**5+18*x**4+27*x**3)*exp(4)-x**6-9*x**5
-27*x**4-27*x**3),x)

[Out]

3*x - 72/(x**4 + x**3*(6 - 2*exp(4)) + x**2*(-6*exp(4) + 9 + exp(8)))

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Giac [A]
time = 0.39, size = 20, normalized size = 0.91 \begin {gather*} 3 \, x - \frac {72}{{\left (x^{2} - x e^{4} + 3 \, x\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+144)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3
-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, alg
orithm="giac")

[Out]

3*x - 72/(x^2 - x*e^4 + 3*x)^2

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Mupad [B]
time = 0.18, size = 42, normalized size = 1.91 \begin {gather*} 3\,x-\frac {\left (18\,{\mathrm {e}}^4-3\,{\mathrm {e}}^8+3\,{\left ({\mathrm {e}}^4-3\right )}^2-27\right )\,x^3+72}{x^2\,{\left (x-{\mathrm {e}}^4+3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((288*x + exp(8)*(27*x^3 + 9*x^4) - 3*x^3*exp(12) - exp(4)*(81*x^3 + 54*x^4 + 9*x^5 + 144) + 81*x^3 + 81*x^
4 + 27*x^5 + 3*x^6 + 432)/(exp(8)*(9*x^3 + 3*x^4) - x^3*exp(12) - exp(4)*(27*x^3 + 18*x^4 + 3*x^5) + 27*x^3 +
27*x^4 + 9*x^5 + x^6),x)

[Out]

3*x - (x^3*(18*exp(4) - 3*exp(8) + 3*(exp(4) - 3)^2 - 27) + 72)/(x^2*(x - exp(4) + 3)^2)

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