3.84.47 \(\int \frac {-9 e^6-162 x+81 x^2}{e^{12} x^2+81 x^4-54 x^5+9 x^6+e^6 (18 x^3-6 x^4)} \, dx\)

Optimal. Leaf size=26 \[ \frac {9}{\left (-3+\frac {e^6}{(-3+x) x}\right ) (-3+x) x^2} \]

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Rubi [B]  time = 0.18, antiderivative size = 57, normalized size of antiderivative = 2.19, number of steps used = 7, number of rules used = 4, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2074, 638, 618, 206} \begin {gather*} \frac {9}{e^6 x}-\frac {27 \left (3 \left (27+4 e^6\right )-\left (27+4 e^6\right ) x\right )}{e^6 \left (27+4 e^6\right ) \left (-3 x^2+9 x+e^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*E^6 - 162*x + 81*x^2)/(E^12*x^2 + 81*x^4 - 54*x^5 + 9*x^6 + E^6*(18*x^3 - 6*x^4)),x]

[Out]

9/(E^6*x) - (27*(3*(27 + 4*E^6) - (27 + 4*E^6)*x))/(E^6*(27 + 4*E^6)*(E^6 + 9*x - 3*x^2))

Rule 206

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

Rule 618

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 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), 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 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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9}{e^6 x^2}+\frac {27 \left (27+2 e^6-9 x\right )}{e^6 \left (e^6+9 x-3 x^2\right )^2}-\frac {27}{e^6 \left (e^6+9 x-3 x^2\right )}\right ) \, dx\\ &=\frac {9}{e^6 x}+\frac {27 \int \frac {27+2 e^6-9 x}{\left (e^6+9 x-3 x^2\right )^2} \, dx}{e^6}-\frac {27 \int \frac {1}{e^6+9 x-3 x^2} \, dx}{e^6}\\ &=\frac {9}{e^6 x}-\frac {27 \left (3 \left (27+4 e^6\right )-\left (27+4 e^6\right ) x\right )}{e^6 \left (27+4 e^6\right ) \left (e^6+9 x-3 x^2\right )}+\frac {27 \int \frac {1}{e^6+9 x-3 x^2} \, dx}{e^6}+\frac {54 \operatorname {Subst}\left (\int \frac {1}{3 \left (27+4 e^6\right )-x^2} \, dx,x,9-6 x\right )}{e^6}\\ &=\frac {9}{e^6 x}-\frac {27 \left (3 \left (27+4 e^6\right )-\left (27+4 e^6\right ) x\right )}{e^6 \left (27+4 e^6\right ) \left (e^6+9 x-3 x^2\right )}+\frac {18 \sqrt {\frac {3}{27+4 e^6}} \tanh ^{-1}\left (\sqrt {\frac {3}{27+4 e^6}} (3-2 x)\right )}{e^6}-\frac {54 \operatorname {Subst}\left (\int \frac {1}{3 \left (27+4 e^6\right )-x^2} \, dx,x,9-6 x\right )}{e^6}\\ &=\frac {9}{e^6 x}-\frac {27 \left (3 \left (27+4 e^6\right )-\left (27+4 e^6\right ) x\right )}{e^6 \left (27+4 e^6\right ) \left (e^6+9 x-3 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 0.73 \begin {gather*} -\frac {9}{x \left (-e^6+3 (-3+x) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*E^6 - 162*x + 81*x^2)/(E^12*x^2 + 81*x^4 - 54*x^5 + 9*x^6 + E^6*(18*x^3 - 6*x^4)),x]

[Out]

-9/(x*(-E^6 + 3*(-3 + x)*x))

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fricas [A]  time = 0.69, size = 20, normalized size = 0.77 \begin {gather*} -\frac {9}{3 \, x^{3} - 9 \, x^{2} - x e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*exp(3)^2+81*x^2-162*x)/(x^2*exp(3)^4+(-6*x^4+18*x^3)*exp(3)^2+9*x^6-54*x^5+81*x^4),x, algorithm=
"fricas")

[Out]

-9/(3*x^3 - 9*x^2 - x*e^6)

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giac [B]  time = 0.34, size = 274, normalized size = 10.54 \begin {gather*} -9 \, {\left (\frac {{\left (3 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - 3\right )}^{2} e^{42} + 36 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - 3\right )} e^{42} + 4 \, e^{48} + 108 \, e^{42}\right )} \log \left (x + \frac {1}{2} \, \sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - \frac {3}{2}\right )}{3 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - 3\right )}^{3} + 27 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - 3\right )}^{2} - 2 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - 3\right )} {\left (2 \, e^{6} - 27\right )} - 12 \, e^{6}} - \frac {{\left (3 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} + 3\right )}^{2} e^{42} - 36 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} + 3\right )} e^{42} + 4 \, e^{48} + 108 \, e^{42}\right )} \log \left (x - \frac {1}{2} \, \sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} - \frac {3}{2}\right )}{3 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} + 3\right )}^{3} - 27 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} + 3\right )}^{2} - 2 \, {\left (\sqrt {\frac {1}{3}} \sqrt {4 \, e^{6} + 27} + 3\right )} {\left (2 \, e^{6} - 27\right )} + 12 \, e^{6}}\right )} e^{\left (-48\right )} + \frac {9 \, e^{\left (-6\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*exp(3)^2+81*x^2-162*x)/(x^2*exp(3)^4+(-6*x^4+18*x^3)*exp(3)^2+9*x^6-54*x^5+81*x^4),x, algorithm=
"giac")

[Out]

-9*((3*(sqrt(1/3)*sqrt(4*e^6 + 27) - 3)^2*e^42 + 36*(sqrt(1/3)*sqrt(4*e^6 + 27) - 3)*e^42 + 4*e^48 + 108*e^42)
*log(x + 1/2*sqrt(1/3)*sqrt(4*e^6 + 27) - 3/2)/(3*(sqrt(1/3)*sqrt(4*e^6 + 27) - 3)^3 + 27*(sqrt(1/3)*sqrt(4*e^
6 + 27) - 3)^2 - 2*(sqrt(1/3)*sqrt(4*e^6 + 27) - 3)*(2*e^6 - 27) - 12*e^6) - (3*(sqrt(1/3)*sqrt(4*e^6 + 27) +
3)^2*e^42 - 36*(sqrt(1/3)*sqrt(4*e^6 + 27) + 3)*e^42 + 4*e^48 + 108*e^42)*log(x - 1/2*sqrt(1/3)*sqrt(4*e^6 + 2
7) - 3/2)/(3*(sqrt(1/3)*sqrt(4*e^6 + 27) + 3)^3 - 27*(sqrt(1/3)*sqrt(4*e^6 + 27) + 3)^2 - 2*(sqrt(1/3)*sqrt(4*
e^6 + 27) + 3)*(2*e^6 - 27) + 12*e^6))*e^(-48) + 9*e^(-6)/x

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maple [A]  time = 0.10, size = 19, normalized size = 0.73

method	result	size
risch	\(\frac {9}{x \left ({\mathrm e}^{6}-3 x^{2}+9 x \right )}\)	\(19\)
gosper	\(\frac {9}{x \left ({\mathrm e}^{6}-3 x^{2}+9 x \right )}\)	\(21\)
norman	\(\frac {9}{x \left ({\mathrm e}^{6}-3 x^{2}+9 x \right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*exp(3)^2+81*x^2-162*x)/(x^2*exp(3)^4+(-6*x^4+18*x^3)*exp(3)^2+9*x^6-54*x^5+81*x^4),x,method=_RETURNVER
BOSE)

[Out]

9/x/(exp(6)-3*x^2+9*x)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.77 \begin {gather*} -\frac {9}{3 \, x^{3} - 9 \, x^{2} - x e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*exp(3)^2+81*x^2-162*x)/(x^2*exp(3)^4+(-6*x^4+18*x^3)*exp(3)^2+9*x^6-54*x^5+81*x^4),x, algorithm=
"maxima")

[Out]

-9/(3*x^3 - 9*x^2 - x*e^6)

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mupad [B]  time = 5.26, size = 18, normalized size = 0.69 \begin {gather*} \frac {9}{x\,\left (-3\,x^2+9\,x+{\mathrm {e}}^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(162*x + 9*exp(6) - 81*x^2)/(exp(6)*(18*x^3 - 6*x^4) + x^2*exp(12) + 81*x^4 - 54*x^5 + 9*x^6),x)

[Out]

9/(x*(9*x + exp(6) - 3*x^2))

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sympy [A]  time = 0.49, size = 17, normalized size = 0.65 \begin {gather*} - \frac {9}{3 x^{3} - 9 x^{2} - x e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*exp(3)**2+81*x**2-162*x)/(x**2*exp(3)**4+(-6*x**4+18*x**3)*exp(3)**2+9*x**6-54*x**5+81*x**4),x)

[Out]

-9/(3*x**3 - 9*x**2 - x*exp(6))

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