3.102.65 \(\int \frac {e^6 (-42-64 x-36 x^2-8 x^3)}{-125-75 x+60 x^2+29 x^3-12 x^4-3 x^5+x^6+e^6 (-45-69 x-23 x^2+8 x^3+4 x^4)} \, dx\) [10165]

Optimal. Leaf size=29 \[ \log \left (1+\frac {e^6 (-3-2 x)^2}{x^2 \left (x-\frac {5+x}{x}\right )^2}\right ) \]

[Out]

ln(1+1/x^2/(x-1/x*(5+x))^2*exp(3)^2*(-2*x-3)^2)

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Rubi [A]
time = 0.17, antiderivative size = 50, normalized size of antiderivative = 1.72, number of steps used = 5, number of rules used = 4, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {12, 2099, 642, 1601} \begin {gather*} \log \left (x^4-2 x^3-\left (9-4 e^6\right ) x^2+2 \left (5+6 e^6\right ) x+9 e^6+25\right )-2 \log \left (-x^2+x+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^6*(-42 - 64*x - 36*x^2 - 8*x^3))/(-125 - 75*x + 60*x^2 + 29*x^3 - 12*x^4 - 3*x^5 + x^6 + E^6*(-45 - 69*
x - 23*x^2 + 8*x^3 + 4*x^4)),x]

[Out]

-2*Log[5 + x - x^2] + Log[25 + 9*E^6 + 2*(5 + 6*E^6)*x - (9 - 4*E^6)*x^2 - 2*x^3 + x^4]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 642

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

Rule 1601

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]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2099

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} &=e^6 \int \frac {-42-64 x-36 x^2-8 x^3}{-125-75 x+60 x^2+29 x^3-12 x^4-3 x^5+x^6+e^6 \left (-45-69 x-23 x^2+8 x^3+4 x^4\right )} \, dx\\ &=e^6 \int \left (-\frac {2 (-1+2 x)}{e^6 \left (-5-x+x^2\right )}+\frac {2 \left (5+6 e^6-\left (9-4 e^6\right ) x-3 x^2+2 x^3\right )}{e^6 \left (25+9 e^6+2 \left (5+6 e^6\right ) x-\left (9-4 e^6\right ) x^2-2 x^3+x^4\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-1+2 x}{-5-x+x^2} \, dx\right )+2 \int \frac {5+6 e^6-\left (9-4 e^6\right ) x-3 x^2+2 x^3}{25+9 e^6+2 \left (5+6 e^6\right ) x-\left (9-4 e^6\right ) x^2-2 x^3+x^4} \, dx\\ &=-2 \log \left (5+x-x^2\right )+\log \left (25+9 e^6+2 \left (5+6 e^6\right ) x-\left (9-4 e^6\right ) x^2-2 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(29)=58\).
time = 0.04, size = 83, normalized size = 2.86 \begin {gather*} -2 e^6 \left (\frac {\log \left (5+8 (3+2 x)-(3+2 x)^2\right )}{e^6}-\frac {\log \left (25+80 (3+2 x)+54 (3+2 x)^2+16 e^6 (3+2 x)^2-16 (3+2 x)^3+(3+2 x)^4\right )}{2 e^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^6*(-42 - 64*x - 36*x^2 - 8*x^3))/(-125 - 75*x + 60*x^2 + 29*x^3 - 12*x^4 - 3*x^5 + x^6 + E^6*(-45
- 69*x - 23*x^2 + 8*x^3 + 4*x^4)),x]

[Out]

-2*E^6*(Log[5 + 8*(3 + 2*x) - (3 + 2*x)^2]/E^6 - Log[25 + 80*(3 + 2*x) + 54*(3 + 2*x)^2 + 16*E^6*(3 + 2*x)^2 -
16*(3 + 2*x)^3 + (3 + 2*x)^4]/(2*E^6))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(30)=60\).
time = 0.06, size = 64, normalized size = 2.21

method	result	size
risch	\(-2 \ln \left (x^{2}-x -5\right )+\ln \left (-x^{4}+2 x^{3}+\left (-4 \,{\mathrm e}^{6}+9\right ) x^{2}+\left (-12 \,{\mathrm e}^{6}-10\right ) x -9 \,{\mathrm e}^{6}-25\right )\)	\(48\)
norman	\(-2 \ln \left (x^{2}-x -5\right )+\ln \left (x^{4}+4 x^{2} {\mathrm e}^{6}-2 x^{3}+12 x \,{\mathrm e}^{6}-9 x^{2}+9 \,{\mathrm e}^{6}+10 x +25\right )\)	\(54\)
default	\(2 \,{\mathrm e}^{6} \left (\frac {{\mathrm e}^{-6} \ln \left (x^{4}+4 x^{2} {\mathrm e}^{6}-2 x^{3}+12 x \,{\mathrm e}^{6}-9 x^{2}+9 \,{\mathrm e}^{6}+10 x +25\right )}{2}-{\mathrm e}^{-6} \ln \left (x^{2}-x -5\right )\right )\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^3-36*x^2-64*x-42)*exp(3)^2/((4*x^4+8*x^3-23*x^2-69*x-45)*exp(3)^2+x^6-3*x^5-12*x^4+29*x^3+60*x^2-75*
x-125),x,method=_RETURNVERBOSE)

[Out]

2*exp(3)^2*(1/2/exp(6)*ln(x^4+4*x^2*exp(6)-2*x^3+12*x*exp(6)-9*x^2+9*exp(6)+10*x+25)-1/exp(6)*ln(x^2-x-5))

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Maxima [A]
time = 0.26, size = 54, normalized size = 1.86 \begin {gather*} {\left (e^{\left (-6\right )} \log \left (x^{4} - 2 \, x^{3} + x^{2} {\left (4 \, e^{6} - 9\right )} + 2 \, x {\left (6 \, e^{6} + 5\right )} + 9 \, e^{6} + 25\right ) - 2 \, e^{\left (-6\right )} \log \left (x^{2} - x - 5\right )\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3-36*x^2-64*x-42)*exp(3)^2/((4*x^4+8*x^3-23*x^2-69*x-45)*exp(3)^2+x^6-3*x^5-12*x^4+29*x^3+60*x
^2-75*x-125),x, algorithm="maxima")

[Out]

(e^(-6)*log(x^4 - 2*x^3 + x^2*(4*e^6 - 9) + 2*x*(6*e^6 + 5) + 9*e^6 + 25) - 2*e^(-6)*log(x^2 - x - 5))*e^6

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Fricas [A]
time = 0.37, size = 44, normalized size = 1.52 \begin {gather*} \log \left (x^{4} - 2 \, x^{3} - 9 \, x^{2} + {\left (4 \, x^{2} + 12 \, x + 9\right )} e^{6} + 10 \, x + 25\right ) - 2 \, \log \left (x^{2} - x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3-36*x^2-64*x-42)*exp(3)^2/((4*x^4+8*x^3-23*x^2-69*x-45)*exp(3)^2+x^6-3*x^5-12*x^4+29*x^3+60*x
^2-75*x-125),x, algorithm="fricas")

[Out]

log(x^4 - 2*x^3 - 9*x^2 + (4*x^2 + 12*x + 9)*e^6 + 10*x + 25) - 2*log(x^2 - x - 5)

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Sympy [A]
time = 1.80, size = 44, normalized size = 1.52 \begin {gather*} - 2 \log {\left (x^{2} - x - 5 \right )} + \log {\left (x^{4} - 2 x^{3} + x^{2} \left (-9 + 4 e^{6}\right ) + x \left (10 + 12 e^{6}\right ) + 25 + 9 e^{6} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**3-36*x**2-64*x-42)*exp(3)**2/((4*x**4+8*x**3-23*x**2-69*x-45)*exp(3)**2+x**6-3*x**5-12*x**4+2
9*x**3+60*x**2-75*x-125),x)

[Out]

-2*log(x**2 - x - 5) + log(x**4 - 2*x**3 + x**2*(-9 + 4*exp(6)) + x*(10 + 12*exp(6)) + 25 + 9*exp(6))

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Giac [A]
time = 0.40, size = 56, normalized size = 1.93 \begin {gather*} {\left (e^{\left (-6\right )} \log \left (x^{4} - 2 \, x^{3} + 4 \, x^{2} e^{6} - 9 \, x^{2} + 12 \, x e^{6} + 10 \, x + 9 \, e^{6} + 25\right ) - 2 \, e^{\left (-6\right )} \log \left ({\left | x^{2} - x - 5 \right |}\right )\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3-36*x^2-64*x-42)*exp(3)^2/((4*x^4+8*x^3-23*x^2-69*x-45)*exp(3)^2+x^6-3*x^5-12*x^4+29*x^3+60*x
^2-75*x-125),x, algorithm="giac")

[Out]

(e^(-6)*log(x^4 - 2*x^3 + 4*x^2*e^6 - 9*x^2 + 12*x*e^6 + 10*x + 9*e^6 + 25) - 2*e^(-6)*log(abs(x^2 - x - 5)))*
e^6

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Mupad [B]
time = 8.39, size = 47, normalized size = 1.62 \begin {gather*} \ln \left (10\,x+9\,{\mathrm {e}}^6+12\,x\,{\mathrm {e}}^6+4\,x^2\,{\mathrm {e}}^6-9\,x^2-2\,x^3+x^4+25\right )-2\,\ln \left (x^2-x-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6)*(64*x + 36*x^2 + 8*x^3 + 42))/(75*x + exp(6)*(69*x + 23*x^2 - 8*x^3 - 4*x^4 + 45) - 60*x^2 - 29*x^
3 + 12*x^4 + 3*x^5 - x^6 + 125),x)

[Out]

log(10*x + 9*exp(6) + 12*x*exp(6) + 4*x^2*exp(6) - 9*x^2 - 2*x^3 + x^4 + 25) - 2*log(x^2 - x - 5)

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