3.64.98 \(\int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 (-120+3 x^4)}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 (120 x+135 x^2+3 x^5)} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {15 \left (9+\frac {8}{x}\right )}{\left (e^4-x\right )^2}+x\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 39, normalized size of antiderivative = 1.86, number of steps used = 5, number of rules used = 3, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6, 2074, 1587} \begin {gather*} \log \left (x^4-2 e^4 x^3+e^8 x^2+135 x+120\right )-2 \log \left (e^4-x\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(360*x + 270*x^2 + E^12*x^2 - 3*E^8*x^3 - x^5 + E^4*(-120 + 3*x^4))/(-120*x^2 - 135*x^3 + E^12*x^3 - 3*E^8
*x^4 - x^6 + E^4*(120*x + 135*x^2 + 3*x^5)),x]

[Out]

-2*Log[E^4 - x] - Log[x] + Log[120 + 135*x + E^8*x^2 - 2*E^4*x^3 + x^4]

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 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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 {gather*} \begin {aligned} \text {integral} &=\int \frac {360 x+\left (270+e^{12}\right ) x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx\\ &=\int \frac {360 x+\left (270+e^{12}\right ) x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2+\left (-135+e^{12}\right ) x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx\\ &=\int \left (\frac {2}{e^4-x}-\frac {1}{x}+\frac {135+2 e^8 x-6 e^4 x^2+4 x^3}{120+135 x+e^8 x^2-2 e^4 x^3+x^4}\right ) \, dx\\ &=-2 \log \left (e^4-x\right )-\log (x)+\int \frac {135+2 e^8 x-6 e^4 x^2+4 x^3}{120+135 x+e^8 x^2-2 e^4 x^3+x^4} \, dx\\ &=-2 \log \left (e^4-x\right )-\log (x)+\log \left (120+135 x+e^8 x^2-2 e^4 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 39, normalized size = 1.86 \begin {gather*} -2 \log \left (e^4-x\right )-\log (x)+\log \left (120+135 x+e^8 x^2-2 e^4 x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(360*x + 270*x^2 + E^12*x^2 - 3*E^8*x^3 - x^5 + E^4*(-120 + 3*x^4))/(-120*x^2 - 135*x^3 + E^12*x^3 -
3*E^8*x^4 - x^6 + E^4*(120*x + 135*x^2 + 3*x^5)),x]

[Out]

-2*Log[E^4 - x] - Log[x] + Log[120 + 135*x + E^8*x^2 - 2*E^4*x^3 + x^4]

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fricas [A]  time = 0.56, size = 36, normalized size = 1.71 \begin {gather*} \log \left (x^{4} - 2 \, x^{3} e^{4} + x^{2} e^{8} + 135 \, x + 120\right ) - 2 \, \log \left (x - e^{4}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3
*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x^3-120*x^2),x, algorithm="fricas")

[Out]

log(x^4 - 2*x^3*e^4 + x^2*e^8 + 135*x + 120) - 2*log(x - e^4) - log(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((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3
*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x^3-120*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.29, size = 39, normalized size = 1.86

method	result	size
risch	\(-\ln \left (-x \right )-2 \ln \left (x -{\mathrm e}^{4}\right )+\ln \left (x^{2} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{4}+x^{4}+135 x +120\right )\)	\(39\)
norman	\(-\ln \relax (x )-2 \ln \left ({\mathrm e}^{4}-x \right )+\ln \left (x^{2} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{4}+x^{4}+135 x +120\right )\)	\(43\)
default	\(-\ln \relax (x )-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}-3 \textit {\_Z}^{4} {\mathrm e}^{4}+3 \textit {\_Z}^{3} {\mathrm e}^{8}+\left (-{\mathrm e}^{12}+135\right ) \textit {\_Z}^{2}+\left (-135 \,{\mathrm e}^{4}+120\right ) \textit {\_Z} -120 \,{\mathrm e}^{4}\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-6 \textit {\_R}^{3} {\mathrm e}^{4}+6 \textit {\_R}^{2} {\mathrm e}^{8}+\left (-2 \,{\mathrm e}^{12}-135\right ) \textit {\_R} -135 \,{\mathrm e}^{4}-240\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}^{12}-9 \textit {\_R}^{2} {\mathrm e}^{8}+12 \textit {\_R}^{3} {\mathrm e}^{4}-5 \textit {\_R}^{4}+135 \,{\mathrm e}^{4}-270 \textit {\_R} -120}\right )\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+1
35*x^2+120*x)*exp(2)^2-x^6-135*x^3-120*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(-x)-2*ln(x-exp(4))+ln(x^2*exp(8)-2*x^3*exp(4)+x^4+135*x+120)

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maxima [A]  time = 0.37, size = 36, normalized size = 1.71 \begin {gather*} \log \left (x^{4} - 2 \, x^{3} e^{4} + x^{2} e^{8} + 135 \, x + 120\right ) - 2 \, \log \left (x - e^{4}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3
*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x^3-120*x^2),x, algorithm="maxima")

[Out]

log(x^4 - 2*x^3*e^4 + x^2*e^8 + 135*x + 120) - 2*log(x - e^4) - log(x)

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mupad [B]  time = 4.37, size = 36, normalized size = 1.71 \begin {gather*} \ln \left (x^4-2\,{\mathrm {e}}^4\,x^3+{\mathrm {e}}^8\,x^2+135\,x+120\right )-2\,\ln \left (x-{\mathrm {e}}^4\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(360*x + exp(4)*(3*x^4 - 120) - 3*x^3*exp(8) + x^2*exp(12) + 270*x^2 - x^5)/(3*x^4*exp(8) - exp(4)*(120*x
+ 135*x^2 + 3*x^5) - x^3*exp(12) + 120*x^2 + 135*x^3 + x^6),x)

[Out]

log(135*x - 2*x^3*exp(4) + x^2*exp(8) + x^4 + 120) - 2*log(x - exp(4)) - log(x)

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sympy [B]  time = 8.06, size = 36, normalized size = 1.71 \begin {gather*} - \log {\relax (x )} - 2 \log {\left (x - e^{4} \right )} + \log {\left (x^{4} - 2 x^{3} e^{4} + x^{2} e^{8} + 135 x + 120 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(2)**6-3*x**3*exp(2)**4+(3*x**4-120)*exp(2)**2-x**5+270*x**2+360*x)/(x**3*exp(2)**6-3*x**4*
exp(2)**4+(3*x**5+135*x**2+120*x)*exp(2)**2-x**6-135*x**3-120*x**2),x)

[Out]

-log(x) - 2*log(x - exp(4)) + log(x**4 - 2*x**3*exp(4) + x**2*exp(8) + 135*x + 120)

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