3.20.9 \(\int \frac {-3600+480 x+164 x^2+e^4 (3600+3120 x+76 x^2-284 x^3+4 x^4)+e^8 (180 x^2+180 x^3+45 x^4)}{900 x^2-120 x^3+4 x^4+e^4 (-1800 x^2-660 x^3+112 x^4-4 x^5)+e^8 (900 x^2+780 x^3+109 x^4-26 x^5+x^6)} \, dx\)

Optimal. Leaf size=35 \[ \frac {2}{15}+\frac {x}{5-\frac {x}{3}}+\frac {4}{x \left (-e^4+\frac {2}{2+x}\right )} \]

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Rubi [A]  time = 0.21, antiderivative size = 53, normalized size of antiderivative = 1.51, number of steps used = 2, number of rules used = 1, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2074} \begin {gather*} \frac {4}{\left (1-e^4\right ) x}+\frac {4}{\left (1-e^4\right ) \left (2 \left (1-e^4\right )-e^4 x\right )}+\frac {45}{15-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3600 + 480*x + 164*x^2 + E^4*(3600 + 3120*x + 76*x^2 - 284*x^3 + 4*x^4) + E^8*(180*x^2 + 180*x^3 + 45*x^
4))/(900*x^2 - 120*x^3 + 4*x^4 + E^4*(-1800*x^2 - 660*x^3 + 112*x^4 - 4*x^5) + E^8*(900*x^2 + 780*x^3 + 109*x^
4 - 26*x^5 + x^6)),x]

[Out]

45/(15 - x) + 4/((1 - E^4)*x) + 4/((1 - E^4)*(2*(1 - E^4) - E^4*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 \left (\frac {45}{(-15+x)^2}+\frac {4}{\left (-1+e^4\right ) x^2}-\frac {4 e^4}{\left (-1+e^4\right ) \left (-2+2 e^4+e^4 x\right )^2}\right ) \, dx\\ &=\frac {45}{15-x}+\frac {4}{\left (1-e^4\right ) x}+\frac {4}{\left (1-e^4\right ) \left (2 \left (1-e^4\right )-e^4 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 46, normalized size = 1.31 \begin {gather*} \frac {120+\left (142-90 e^4\right ) x-\left (4+45 e^4\right ) x^2}{x \left (30-2 x+e^4 \left (-30-13 x+x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3600 + 480*x + 164*x^2 + E^4*(3600 + 3120*x + 76*x^2 - 284*x^3 + 4*x^4) + E^8*(180*x^2 + 180*x^3 +
45*x^4))/(900*x^2 - 120*x^3 + 4*x^4 + E^4*(-1800*x^2 - 660*x^3 + 112*x^4 - 4*x^5) + E^8*(900*x^2 + 780*x^3 +
109*x^4 - 26*x^5 + x^6)),x]

[Out]

(120 + (142 - 90*E^4)*x - (4 + 45*E^4)*x^2)/(x*(30 - 2*x + E^4*(-30 - 13*x + x^2)))

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fricas [A]  time = 0.65, size = 49, normalized size = 1.40 \begin {gather*} \frac {4 \, x^{2} + 45 \, {\left (x^{2} + 2 \, x\right )} e^{4} - 142 \, x - 120}{2 \, x^{2} - {\left (x^{3} - 13 \, x^{2} - 30 \, x\right )} e^{4} - 30 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x^4+180*x^3+180*x^2)*exp(4)^2+(4*x^4-284*x^3+76*x^2+3120*x+3600)*exp(4)+164*x^2+480*x-3600)/((x
^6-26*x^5+109*x^4+780*x^3+900*x^2)*exp(4)^2+(-4*x^5+112*x^4-660*x^3-1800*x^2)*exp(4)+4*x^4-120*x^3+900*x^2),x,
algorithm="fricas")

[Out]

(4*x^2 + 45*(x^2 + 2*x)*e^4 - 142*x - 120)/(2*x^2 - (x^3 - 13*x^2 - 30*x)*e^4 - 30*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x^4+180*x^3+180*x^2)*exp(4)^2+(4*x^4-284*x^3+76*x^2+3120*x+3600)*exp(4)+164*x^2+480*x-3600)/((x
^6-26*x^5+109*x^4+780*x^3+900*x^2)*exp(4)^2+(-4*x^5+112*x^4-660*x^3-1800*x^2)*exp(4)+4*x^4-120*x^3+900*x^2),x,
algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-2*exp(8)+2*exp(4)^2)/(exp(8)^2-4*exp(8
)*exp(4)+2*exp(8)+4*exp(4)^2-4*exp(4)+1)*ln(sageVARx^2*exp(8)+4*sageVARx*exp(8)-4*sageVARx*exp(4)+4*exp(8)-8*e
xp(4)+4)+(-4*exp(8)^2

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maple [A]  time = 55.98, size = 47, normalized size = 1.34

method	result	size
risch	\(\frac {\left (-45 \,{\mathrm e}^{4}-4\right ) x^{2}+\left (-90 \,{\mathrm e}^{4}+142\right ) x +120}{\left (x^{2} {\mathrm e}^{4}-13 x \,{\mathrm e}^{4}-30 \,{\mathrm e}^{4}-2 x +30\right ) x}\)	\(47\)
gosper	\(-\frac {45 x^{2} {\mathrm e}^{4}+90 x \,{\mathrm e}^{4}+4 x^{2}-142 x -120}{x \left (x^{2} {\mathrm e}^{4}-13 x \,{\mathrm e}^{4}-30 \,{\mathrm e}^{4}-2 x +30\right )}\)	\(50\)
norman	\(\frac {120-\left (45 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4} x^{2}-2 \left (45 \,{\mathrm e}^{8}-71 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4} x}{x \left (x -15\right ) \left (x \,{\mathrm e}^{4}+2 \,{\mathrm e}^{4}-2\right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((45*x^4+180*x^3+180*x^2)*exp(4)^2+(4*x^4-284*x^3+76*x^2+3120*x+3600)*exp(4)+164*x^2+480*x-3600)/((x^6-26*
x^5+109*x^4+780*x^3+900*x^2)*exp(4)^2+(-4*x^5+112*x^4-660*x^3-1800*x^2)*exp(4)+4*x^4-120*x^3+900*x^2),x,method
=_RETURNVERBOSE)

[Out]

((-45*exp(4)-4)*x^2+(-90*exp(4)+142)*x+120)/(x^2*exp(4)-13*x*exp(4)-30*exp(4)-2*x+30)/x

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maxima [A]  time = 0.59, size = 50, normalized size = 1.43 \begin {gather*} -\frac {x^{2} {\left (45 \, e^{4} + 4\right )} + 2 \, x {\left (45 \, e^{4} - 71\right )} - 120}{x^{3} e^{4} - x^{2} {\left (13 \, e^{4} + 2\right )} - 30 \, x {\left (e^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x^4+180*x^3+180*x^2)*exp(4)^2+(4*x^4-284*x^3+76*x^2+3120*x+3600)*exp(4)+164*x^2+480*x-3600)/((x
^6-26*x^5+109*x^4+780*x^3+900*x^2)*exp(4)^2+(-4*x^5+112*x^4-660*x^3-1800*x^2)*exp(4)+4*x^4-120*x^3+900*x^2),x,
algorithm="maxima")

[Out]

-(x^2*(45*e^4 + 4) + 2*x*(45*e^4 - 71) - 120)/(x^3*e^4 - x^2*(13*e^4 + 2) - 30*x*(e^4 - 1))

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mupad [B]  time = 1.35, size = 39, normalized size = 1.11 \begin {gather*} \frac {4}{\left ({\mathrm {e}}^4-1\right )\,\left (2\,{\mathrm {e}}^4+x\,{\mathrm {e}}^4-2\right )}-\frac {45}{x-15}-\frac {4}{x\,\left ({\mathrm {e}}^4-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((480*x + exp(4)*(3120*x + 76*x^2 - 284*x^3 + 4*x^4 + 3600) + exp(8)*(180*x^2 + 180*x^3 + 45*x^4) + 164*x^2
- 3600)/(exp(8)*(900*x^2 + 780*x^3 + 109*x^4 - 26*x^5 + x^6) + 900*x^2 - 120*x^3 + 4*x^4 - exp(4)*(1800*x^2 +
660*x^3 - 112*x^4 + 4*x^5)),x)

[Out]

4/((exp(4) - 1)*(2*exp(4) + x*exp(4) - 2)) - 45/(x - 15) - 4/(x*(exp(4) - 1))

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sympy [B]  time = 2.84, size = 48, normalized size = 1.37 \begin {gather*} \frac {x^{2} \left (- 45 e^{4} - 4\right ) + x \left (142 - 90 e^{4}\right ) + 120}{x^{3} e^{4} + x^{2} \left (- 13 e^{4} - 2\right ) + x \left (30 - 30 e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x**4+180*x**3+180*x**2)*exp(4)**2+(4*x**4-284*x**3+76*x**2+3120*x+3600)*exp(4)+164*x**2+480*x-3
600)/((x**6-26*x**5+109*x**4+780*x**3+900*x**2)*exp(4)**2+(-4*x**5+112*x**4-660*x**3-1800*x**2)*exp(4)+4*x**4-
120*x**3+900*x**2),x)

[Out]

(x**2*(-45*exp(4) - 4) + x*(142 - 90*exp(4)) + 120)/(x**3*exp(4) + x**2*(-13*exp(4) - 2) + x*(30 - 30*exp(4)))

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