3.99.38 \(\int \frac {144-144 x+38 x^2+e^4 (12-12 x+3 x^2)}{144 x-128 x^2+18 x^3+5 x^4+e^4 (12 x-12 x^2+3 x^3)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {2}{4-\frac {3 \left (-12-e^4\right )}{x}+\frac {x}{-2+x}}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 2, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2074, 628} \begin {gather*} -\log \left (-5 x^2-\left (28+3 e^4\right ) x+6 \left (12+e^4\right )\right )+\log (2-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(144 - 144*x + 38*x^2 + E^4*(12 - 12*x + 3*x^2))/(144*x - 128*x^2 + 18*x^3 + 5*x^4 + E^4*(12*x - 12*x^2 +
3*x^3)),x]

[Out]

Log[2 - x] + Log[x] - Log[6*(12 + E^4) - (28 + 3*E^4)*x - 5*x^2]

Rule 628

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 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 {1}{-2+x}+\frac {1}{x}+\frac {28+3 e^4+10 x}{6 \left (12+e^4\right )-\left (28+3 e^4\right ) x-5 x^2}\right ) \, dx\\ &=\log (2-x)+\log (x)+\int \frac {28+3 e^4+10 x}{6 \left (12+e^4\right )+\left (-28-3 e^4\right ) x-5 x^2} \, dx\\ &=\log (2-x)+\log (x)-\log \left (6 \left (12+e^4\right )-\left (28+3 e^4\right ) x-5 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 33, normalized size = 1.27 \begin {gather*} \log (2-x)+\log (x)-\log \left (72+6 e^4-28 x-3 e^4 x-5 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(144 - 144*x + 38*x^2 + E^4*(12 - 12*x + 3*x^2))/(144*x - 128*x^2 + 18*x^3 + 5*x^4 + E^4*(12*x - 12*
x^2 + 3*x^3)),x]

[Out]

Log[2 - x] + Log[x] - Log[72 + 6*E^4 - 28*x - 3*E^4*x - 5*x^2]

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fricas [A]  time = 0.52, size = 29, normalized size = 1.12 \begin {gather*} -\log \left (5 \, x^{2} + 3 \, {\left (x - 2\right )} e^{4} + 28 \, x - 72\right ) + \log \left (x^{2} - 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-12*x+12)*exp(2)^2+38*x^2-144*x+144)/((3*x^3-12*x^2+12*x)*exp(2)^2+5*x^4+18*x^3-128*x^2+144*x
),x, algorithm="fricas")

[Out]

-log(5*x^2 + 3*(x - 2)*e^4 + 28*x - 72) + log(x^2 - 2*x)

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giac [A]  time = 0.22, size = 32, normalized size = 1.23 \begin {gather*} -\log \left ({\left | 5 \, x^{2} + 3 \, x e^{4} + 28 \, x - 6 \, e^{4} - 72 \right |}\right ) + \log \left ({\left | x - 2 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-12*x+12)*exp(2)^2+38*x^2-144*x+144)/((3*x^3-12*x^2+12*x)*exp(2)^2+5*x^4+18*x^3-128*x^2+144*x
),x, algorithm="giac")

[Out]

-log(abs(5*x^2 + 3*x*e^4 + 28*x - 6*e^4 - 72)) + log(abs(x - 2)) + log(abs(x))

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maple [A]  time = 0.08, size = 30, normalized size = 1.15

method	result	size
default	\(\ln \relax (x )+\ln \left (x -2\right )-\ln \left (3 x \,{\mathrm e}^{4}+5 x^{2}-6 \,{\mathrm e}^{4}+28 x -72\right )\)	\(30\)
norman	\(\ln \relax (x )+\ln \left (x -2\right )-\ln \left (3 x \,{\mathrm e}^{4}+5 x^{2}-6 \,{\mathrm e}^{4}+28 x -72\right )\)	\(34\)
risch	\(-\ln \left (-5 x^{2}+\left (-3 \,{\mathrm e}^{4}-28\right ) x +6 \,{\mathrm e}^{4}+72\right )+\ln \left (-x^{2}+2 x \right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2-12*x+12)*exp(2)^2+38*x^2-144*x+144)/((3*x^3-12*x^2+12*x)*exp(2)^2+5*x^4+18*x^3-128*x^2+144*x),x,me
thod=_RETURNVERBOSE)

[Out]

ln(x)+ln(x-2)-ln(3*x*exp(4)+5*x^2-6*exp(4)+28*x-72)

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maxima [A]  time = 0.35, size = 29, normalized size = 1.12 \begin {gather*} -\log \left (5 \, x^{2} + x {\left (3 \, e^{4} + 28\right )} - 6 \, e^{4} - 72\right ) + \log \left (x - 2\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-12*x+12)*exp(2)^2+38*x^2-144*x+144)/((3*x^3-12*x^2+12*x)*exp(2)^2+5*x^4+18*x^3-128*x^2+144*x
),x, algorithm="maxima")

[Out]

-log(5*x^2 + x*(3*e^4 + 28) - 6*e^4 - 72) + log(x - 2) + log(x)

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mupad [B]  time = 5.94, size = 91, normalized size = 3.50 \begin {gather*} \mathrm {atan}\left (\frac {-x\,5008{}\mathrm {i}+{\mathrm {e}}^4\,120{}\mathrm {i}-x\,{\mathrm {e}}^4\,636{}\mathrm {i}-x\,{\mathrm {e}}^8\,18{}\mathrm {i}+x^2\,{\mathrm {e}}^4\,288{}\mathrm {i}+x^2\,{\mathrm {e}}^8\,9{}\mathrm {i}+x^2\,2124{}\mathrm {i}+1440{}\mathrm {i}}{3888\,x+120\,{\mathrm {e}}^4+516\,x\,{\mathrm {e}}^4+18\,x\,{\mathrm {e}}^8-288\,x^2\,{\mathrm {e}}^4-9\,x^2\,{\mathrm {e}}^8-2324\,x^2+1440}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*(3*x^2 - 12*x + 12) - 144*x + 38*x^2 + 144)/(144*x + exp(4)*(12*x - 12*x^2 + 3*x^3) - 128*x^2 + 18
*x^3 + 5*x^4),x)

[Out]

atan((exp(4)*120i - x*5008i - x*exp(4)*636i - x*exp(8)*18i + x^2*exp(4)*288i + x^2*exp(8)*9i + x^2*2124i + 144
0i)/(3888*x + 120*exp(4) + 516*x*exp(4) + 18*x*exp(8) - 288*x^2*exp(4) - 9*x^2*exp(8) - 2324*x^2 + 1440))*2i

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sympy [A]  time = 1.46, size = 34, normalized size = 1.31 \begin {gather*} \log {\left (x^{2} - 2 x \right )} - \log {\left (x^{2} + x \left (\frac {28}{5} + \frac {3 e^{4}}{5}\right ) - \frac {6 e^{4}}{5} - \frac {72}{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2-12*x+12)*exp(2)**2+38*x**2-144*x+144)/((3*x**3-12*x**2+12*x)*exp(2)**2+5*x**4+18*x**3-128*x
**2+144*x),x)

[Out]

log(x**2 - 2*x) - log(x**2 + x*(28/5 + 3*exp(4)/5) - 6*exp(4)/5 - 72/5)

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