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

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

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Rubi [B]  time = 0.18, 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, 2074} $$\begin{array}{c}-\frac{72}{{(3-e^4)}^2{x}^2}+3x-\frac{144}{{(3-e^4)}^3(x-e^4+3)}-\frac{72}{{(3-e^4)}^2{(x-e^4+3)}^2}+\frac{144}{{(3-e^4)}^{3}x}\end{array}$$

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 2074

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-432-288x-81x^3+3e^{12}{x}^3-81x^4-27x^5-3x^6+e^8(-27x^3-9x^4)+e^4(144+81x^3+54x^4+9x^5)}{(-27+e^{12})x^3-27x^4-9x^5-x^6+e^8(-9x^3-3x^4)+e^4(27x^3+18x^4+3x^5)}dx\\ & =\int \frac{-432-288x+(-81+3e^{12})x^3-81x^4-27x^5-3x^6+e^8(-27x^3-9x^4)+e^4(144+81x^3+54x^4+9x^5)}{(-27+e^{12})x^3-27x^4-9x^5-x^6+e^8(-9x^3-3x^4)+e^4(27x^3+18x^4+3x^5)}dx\\ & =\int (3-\frac{144}{{(-3+e^4)}^2{(-3+e^4-x)}^3}-\frac{144}{{(-3+e^4)}^3{(-3+e^4-x)}^2}+\frac{144}{{(-3+e^4)}^2{x}^3}+\frac{144}{{(-3+e^4)}^3{x}^2})dx\\ & =-\frac{72}{{(3-e^4)}^2{x}^2}+\frac{144}{{(3-e^4)}^{3}x}+3x-\frac{72}{{(3-e^4)}^2{(3-e^4+x)}^2}-\frac{144}{{(3-e^4)}^3(3-e^4+x)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.04, size = 41, normalized size = 1.86 $$\begin{array}{c}\frac{3(-24+{(-3+e^4)}^2{x}^3-2(-3+e^4)x^4+x^5)}{x^2{(3-e^4+x)}^2}\end{array}$$

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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fricas [B]  time = 0.67, size = 71, normalized size = 3.23 $$\begin{array}{c}\frac{3(x^5+6x^4+x^3{e}^8+9x^3-2(x^4+3x^3)e^4-24)}{x^4+6x^3+x^2{e}^8+9x^2-2(x^3+3x^2)e^4}\end{array}$$

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

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]

Timed out

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maple [A]  time = 0.17, size = 31, normalized size = 1.41

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

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

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maxima [A]  time = 0.40, size = 33, normalized size = 1.50 $$\begin{array}{c}3x-\frac{72}{x^4-2x^3(e^4-3)+x^2(e^8-6e^4+9)}\end{array}$$

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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mupad [B]  time = 0.18, size = 42, normalized size = 1.91 $$\begin{array}{c}3x-\frac{(18{\mathrm{e}}^4-3{\mathrm{e}}^8+3{({\mathrm{e}}^4-3)}^2-27)x^3+72}{x^2{(x-{\mathrm{e}}^4+3)}^2}\end{array}$$

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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sympy [A]  time = 0.84, size = 31, normalized size = 1.41 $$\begin{array}{c}3x-\frac{72}{x^4+x^3(6-2e^4)+x^2(-6e^4+9+e^8)}\end{array}$$

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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