3.38.20 \(\int \frac {4 e^6 x+2 x^3-4 x^5+2 x^7+e^2 (1-3 x^2)+e^4 (-2 x^2-6 x^6)}{x^2-2 x^4+4 e^8 x^4+x^6+e^4 (4 x^3-4 x^5)} \, dx\)

Optimal. Leaf size=31 \[ \frac {1-\frac {e^2}{x}-x^4}{1+2 e^4 x-x^2} \]

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Rubi [B]  time = 0.23, antiderivative size = 65, normalized size of antiderivative = 2.10, number of steps used = 8, number of rules used = 5, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6, 2074, 638, 618, 206} \begin {gather*} x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{-x^2+2 e^4 x+1}+2 e^4 x-\frac {e^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^6*x + 2*x^3 - 4*x^5 + 2*x^7 + E^2*(1 - 3*x^2) + E^4*(-2*x^2 - 6*x^6))/(x^2 - 2*x^4 + 4*E^8*x^4 + x^6
+ E^4*(4*x^3 - 4*x^5)),x]

[Out]

-(E^2/x) + 2*E^4*x + x^2 + (E^2*(2*E^4*(1 - 2*E^2) - (1 + 4*E^2 + 8*E^10)*x))/(1 + 2*E^4*x - 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 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 \frac {4 e^6 x+2 x^3-4 x^5+2 x^7+e^2 \left (1-3 x^2\right )+e^4 \left (-2 x^2-6 x^6\right )}{x^2+\left (-2+4 e^8\right ) x^4+x^6+e^4 \left (4 x^3-4 x^5\right )} \, dx\\ &=\int \left (2 e^4+\frac {e^2}{x^2}+2 x+\frac {2 \left (-e^2 \left (1+4 e^2+2 e^8+4 e^{10}\right )+e^6 \left (1-8 e^2-8 e^{10}\right ) x\right )}{\left (1+2 e^4 x-x^2\right )^2}+\frac {e^2+4 e^4+8 e^{12}}{1+2 e^4 x-x^2}\right ) \, dx\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+2 \int \frac {-e^2 \left (1+4 e^2+2 e^8+4 e^{10}\right )+e^6 \left (1-8 e^2-8 e^{10}\right ) x}{\left (1+2 e^4 x-x^2\right )^2} \, dx+\left (e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \int \frac {1}{1+2 e^4 x-x^2} \, dx\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}-\left (e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \int \frac {1}{1+2 e^4 x-x^2} \, dx-\left (2 e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+e^8\right )-x^2} \, dx,x,2 e^4-2 x\right )\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}-\frac {e^2 \left (1+4 e^2+8 e^{10}\right ) \tanh ^{-1}\left (\frac {e^4-x}{\sqrt {1+e^8}}\right )}{\sqrt {1+e^8}}+\left (2 e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+e^8\right )-x^2} \, dx,x,2 e^4-2 x\right )\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 57, normalized size = 1.84 \begin {gather*} \frac {e^2+2 e^4 x^2+8 e^{12} x^2-4 e^8 x \left (-1+x^2\right )+x^3 \left (-1+x^2\right )}{x \left (-1-2 e^4 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^6*x + 2*x^3 - 4*x^5 + 2*x^7 + E^2*(1 - 3*x^2) + E^4*(-2*x^2 - 6*x^6))/(x^2 - 2*x^4 + 4*E^8*x^4
+ x^6 + E^4*(4*x^3 - 4*x^5)),x]

[Out]

(E^2 + 2*E^4*x^2 + 8*E^12*x^2 - 4*E^8*x*(-1 + x^2) + x^3*(-1 + x^2))/(x*(-1 - 2*E^4*x + x^2))

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fricas [B]  time = 0.63, size = 53, normalized size = 1.71 \begin {gather*} \frac {x^{5} - x^{3} + 8 \, x^{2} e^{12} + 2 \, x^{2} e^{4} - 4 \, {\left (x^{3} - x\right )} e^{8} + e^{2}}{x^{3} - 2 \, x^{2} e^{4} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(2)^3+(-6*x^6-2*x^2)*exp(2)^2+(-3*x^2+1)*exp(2)+2*x^7-4*x^5+2*x^3)/(4*x^4*exp(2)^4+(-4*x^5+4
*x^3)*exp(2)^2+x^6-2*x^4+x^2),x, algorithm="fricas")

[Out]

(x^5 - x^3 + 8*x^2*e^12 + 2*x^2*e^4 - 4*(x^3 - x)*e^8 + e^2)/(x^3 - 2*x^2*e^4 - x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(2)^3+(-6*x^6-2*x^2)*exp(2)^2+(-3*x^2+1)*exp(2)+2*x^7-4*x^5+2*x^3)/(4*x^4*exp(2)^4+(-4*x^5+4
*x^3)*exp(2)^2+x^6-2*x^4+x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.15, size = 31, normalized size = 1.00

method	result	size
gosper	\(-\frac {x^{5}+{\mathrm e}^{2}-x}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\)	\(31\)
norman	\(\frac {-x^{5}+x -{\mathrm e}^{2}}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\)	\(32\)
risch	\(2 x \,{\mathrm e}^{4}+x^{2}+\frac {-4 \left (2 \,{\mathrm e}^{8}+1\right ) {\mathrm e}^{4} x^{2}-4 x \,{\mathrm e}^{8}-{\mathrm e}^{2}}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\)	\(52\)
default	\(2 x \,{\mathrm e}^{4}+x^{2}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\textit {\_Z}^{4}-4 \textit {\_Z}^{3} {\mathrm e}^{4}+\left (4 \,{\mathrm e}^{8}-2\right ) \textit {\_Z}^{2}+4 \textit {\_Z} \,{\mathrm e}^{4}\right )}{\sum }\frac {\left (\left (-8 \,{\mathrm e}^{12}-{\mathrm e}^{2}-4 \,{\mathrm e}^{4}\right ) \textit {\_R}^{2}+4 \left (-2 \,{\mathrm e}^{8}+{\mathrm e}^{6}\right ) \textit {\_R} -{\mathrm e}^{2}-4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{10}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}^{8}-3 \textit {\_R}^{2} {\mathrm e}^{4}+\textit {\_R}^{3}+{\mathrm e}^{4}-\textit {\_R}}\right )}{4}-\frac {{\mathrm e}^{2}}{x}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(2)^3+(-6*x^6-2*x^2)*exp(2)^2+(-3*x^2+1)*exp(2)+2*x^7-4*x^5+2*x^3)/(4*x^4*exp(2)^4+(-4*x^5+4*x^3)*
exp(2)^2+x^6-2*x^4+x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 46, normalized size = 1.48 \begin {gather*} x^{2} + 2 \, x e^{4} + \frac {4 \, x^{2} {\left (2 \, e^{12} + e^{4}\right )} + 4 \, x e^{8} + e^{2}}{x^{3} - 2 \, x^{2} e^{4} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(2)^3+(-6*x^6-2*x^2)*exp(2)^2+(-3*x^2+1)*exp(2)+2*x^7-4*x^5+2*x^3)/(4*x^4*exp(2)^4+(-4*x^5+4
*x^3)*exp(2)^2+x^6-2*x^4+x^2),x, algorithm="maxima")

[Out]

x^2 + 2*x*e^4 + (4*x^2*(2*e^12 + e^4) + 4*x*e^8 + e^2)/(x^3 - 2*x^2*e^4 - x)

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mupad [B]  time = 2.38, size = 48, normalized size = 1.55 \begin {gather*} 2\,x\,{\mathrm {e}}^4-\frac {\left (4\,{\mathrm {e}}^4+8\,{\mathrm {e}}^{12}\right )\,x^2+4\,{\mathrm {e}}^8\,x+{\mathrm {e}}^2}{-x^3+2\,{\mathrm {e}}^4\,x^2+x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(6) - exp(2)*(3*x^2 - 1) - exp(4)*(2*x^2 + 6*x^6) + 2*x^3 - 4*x^5 + 2*x^7)/(exp(4)*(4*x^3 - 4*x^5)
+ 4*x^4*exp(8) + x^2 - 2*x^4 + x^6),x)

[Out]

2*x*exp(4) - (exp(2) + x^2*(4*exp(4) + 8*exp(12)) + 4*x*exp(8))/(x + 2*x^2*exp(4) - x^3) + x^2

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sympy [B]  time = 4.86, size = 46, normalized size = 1.48 \begin {gather*} x^{2} + 2 x e^{4} + \frac {x^{2} \left (4 e^{4} + 8 e^{12}\right ) + 4 x e^{8} + e^{2}}{x^{3} - 2 x^{2} e^{4} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(2)**3+(-6*x**6-2*x**2)*exp(2)**2+(-3*x**2+1)*exp(2)+2*x**7-4*x**5+2*x**3)/(4*x**4*exp(2)**4
+(-4*x**5+4*x**3)*exp(2)**2+x**6-2*x**4+x**2),x)

[Out]

x**2 + 2*x*exp(4) + (x**2*(4*exp(4) + 8*exp(12)) + 4*x*exp(8) + exp(2))/(x**3 - 2*x**2*exp(4) - x)

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