3.89.92 \(\int \frac {8 x^5+e^8 (-48+8 x+16 x^2+2 x^3)+e^4 (192 x-80 x^3-8 x^4)}{-8 x^4+4 x^6+e^8 (-8+8 x+2 x^2-4 x^3+x^4)+e^4 (-16 x^2+8 x^3+8 x^4-4 x^5)} \, dx\)

Optimal. Leaf size=29 \[ \frac {12}{-1+\frac {2}{x-\frac {2 x^2}{e^4}}}+\log \left (2-x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 33, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2074, 629, 260} \begin {gather*} \frac {24 e^4}{2 x^2-e^4 x+2 e^4}+\log \left (2-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*x^5 + E^8*(-48 + 8*x + 16*x^2 + 2*x^3) + E^4*(192*x - 80*x^3 - 8*x^4))/(-8*x^4 + 4*x^6 + E^8*(-8 + 8*x
+ 2*x^2 - 4*x^3 + x^4) + E^4*(-16*x^2 + 8*x^3 + 8*x^4 - 4*x^5)),x]

[Out]

(24*E^4)/(2*E^4 - E^4*x + 2*x^2) + Log[2 - x^2]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

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 {24 \left (e^8-4 e^4 x\right )}{\left (-2 e^4+e^4 x-2 x^2\right )^2}+\frac {2 x}{-2+x^2}\right ) \, dx\\ &=2 \int \frac {x}{-2+x^2} \, dx+24 \int \frac {e^8-4 e^4 x}{\left (-2 e^4+e^4 x-2 x^2\right )^2} \, dx\\ &=\frac {24 e^4}{2 e^4-e^4 x+2 x^2}+\log \left (2-x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 29, normalized size = 1.00 \begin {gather*} -\frac {24 e^4}{e^4 (-2+x)-2 x^2}+\log \left (2-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^5 + E^8*(-48 + 8*x + 16*x^2 + 2*x^3) + E^4*(192*x - 80*x^3 - 8*x^4))/(-8*x^4 + 4*x^6 + E^8*(-8
+ 8*x + 2*x^2 - 4*x^3 + x^4) + E^4*(-16*x^2 + 8*x^3 + 8*x^4 - 4*x^5)),x]

[Out]

(-24*E^4)/(E^4*(-2 + x) - 2*x^2) + Log[2 - x^2]

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 41, normalized size = 1.41 \begin {gather*} \frac {{\left (2 \, x^{2} - {\left (x - 2\right )} e^{4}\right )} \log \left (x^{2} - 2\right ) + 24 \, e^{4}}{2 \, x^{2} - {\left (x - 2\right )} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+16*x^2+8*x-48)*exp(4)^2+(-8*x^4-80*x^3+192*x)*exp(4)+8*x^5)/((x^4-4*x^3+2*x^2+8*x-8)*exp(4)^
2+(-4*x^5+8*x^4+8*x^3-16*x^2)*exp(4)+4*x^6-8*x^4),x, algorithm="fricas")

[Out]

((2*x^2 - (x - 2)*e^4)*log(x^2 - 2) + 24*e^4)/(2*x^2 - (x - 2)*e^4)

________________________________________________________________________________________

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(((2*x^3+16*x^2+8*x-48)*exp(4)^2+(-8*x^4-80*x^3+192*x)*exp(4)+8*x^5)/((x^4-4*x^3+2*x^2+8*x-8)*exp(4)^
2+(-4*x^5+8*x^4+8*x^3-16*x^2)*exp(4)+4*x^6-8*x^4),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.24, size = 28, normalized size = 0.97

method	result	size
norman	\(-\frac {24 \,{\mathrm e}^{4}}{x \,{\mathrm e}^{4}-2 x^{2}-2 \,{\mathrm e}^{4}}+\ln \left (x^{2}-2\right )\)	\(28\)
risch	\(-\frac {24 \,{\mathrm e}^{4}}{x \,{\mathrm e}^{4}-2 x^{2}-2 \,{\mathrm e}^{4}}+\ln \left (x^{2}-2\right )\)	\(28\)
default	\(12 \left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{4}-4 \textit {\_Z}^{3} {\mathrm e}^{4}+\left ({\mathrm e}^{8}+8 \,{\mathrm e}^{4}\right ) \textit {\_Z}^{2}-4 \textit {\_Z} \,{\mathrm e}^{8}+4 \,{\mathrm e}^{8}\right )}{\sum }\frac {\left (-4 \textit {\_R} \,{\mathrm e}^{4}+{\mathrm e}^{8}\right ) \ln \left (x -\textit {\_R} \right )}{-6 \textit {\_R}^{2} {\mathrm e}^{4}+8 \textit {\_R}^{3}+\textit {\_R} \,{\mathrm e}^{8}+8 \textit {\_R} \,{\mathrm e}^{4}-2 \,{\mathrm e}^{8}}\right )+\ln \left (x^{2}-2\right )\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+16*x^2+8*x-48)*exp(4)^2+(-8*x^4-80*x^3+192*x)*exp(4)+8*x^5)/((x^4-4*x^3+2*x^2+8*x-8)*exp(4)^2+(-4*
x^5+8*x^4+8*x^3-16*x^2)*exp(4)+4*x^6-8*x^4),x,method=_RETURNVERBOSE)

[Out]

-24*exp(4)/(x*exp(4)-2*x^2-2*exp(4))+ln(x^2-2)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 28, normalized size = 0.97 \begin {gather*} \frac {24 \, e^{4}}{2 \, x^{2} - x e^{4} + 2 \, e^{4}} + \log \left (x^{2} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+16*x^2+8*x-48)*exp(4)^2+(-8*x^4-80*x^3+192*x)*exp(4)+8*x^5)/((x^4-4*x^3+2*x^2+8*x-8)*exp(4)^
2+(-4*x^5+8*x^4+8*x^3-16*x^2)*exp(4)+4*x^6-8*x^4),x, algorithm="maxima")

[Out]

24*e^4/(2*x^2 - x*e^4 + 2*e^4) + log(x^2 - 2)

________________________________________________________________________________________

mupad [B]  time = 0.26, size = 28, normalized size = 0.97 \begin {gather*} \ln \left (x^2-2\right )+\frac {24\,{\mathrm {e}}^4}{2\,x^2-{\mathrm {e}}^4\,x+2\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8)*(8*x + 16*x^2 + 2*x^3 - 48) - exp(4)*(80*x^3 - 192*x + 8*x^4) + 8*x^5)/(exp(8)*(8*x + 2*x^2 - 4*x^
3 + x^4 - 8) - 8*x^4 + 4*x^6 - exp(4)*(16*x^2 - 8*x^3 - 8*x^4 + 4*x^5)),x)

[Out]

log(x^2 - 2) + (24*exp(4))/(2*exp(4) - x*exp(4) + 2*x^2)

________________________________________________________________________________________

sympy [A]  time = 2.18, size = 26, normalized size = 0.90 \begin {gather*} \log {\left (x^{2} - 2 \right )} + \frac {24 e^{4}}{2 x^{2} - x e^{4} + 2 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+16*x**2+8*x-48)*exp(4)**2+(-8*x**4-80*x**3+192*x)*exp(4)+8*x**5)/((x**4-4*x**3+2*x**2+8*x-8
)*exp(4)**2+(-4*x**5+8*x**4+8*x**3-16*x**2)*exp(4)+4*x**6-8*x**4),x)

[Out]

log(x**2 - 2) + 24*exp(4)/(2*x**2 - x*exp(4) + 2*exp(4))

________________________________________________________________________________________