3.64.60 \(\int \frac {5634+1171875 e^4+22500 x+14062500 x^2+e^4 (1175625+4687500 x) \log (\frac {1}{x})+390625 e^8 \log ^2(\frac {1}{x})}{9+22500 x+14062500 x^2+e^4 (3750+4687500 x) \log (\frac {1}{x})+390625 e^8 \log ^2(\frac {1}{x})} \, dx\)

Optimal. Leaf size=25 \[ -1+x+\frac {x}{\frac {1}{625}+2 x+\frac {1}{3} e^4 \log \left (\frac {1}{x}\right )} \]

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Rubi [F]  time = 0.36, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {5634+1171875 e^4+22500 x+14062500 x^2+e^4 (1175625+4687500 x) \log \left (\frac {1}{x}\right )+390625 e^8 \log ^2\left (\frac {1}{x}\right )}{9+22500 x+14062500 x^2+e^4 (3750+4687500 x) \log \left (\frac {1}{x}\right )+390625 e^8 \log ^2\left (\frac {1}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5634 + 1171875*E^4 + 22500*x + 14062500*x^2 + E^4*(1175625 + 4687500*x)*Log[x^(-1)] + 390625*E^8*Log[x^(-
1)]^2)/(9 + 22500*x + 14062500*x^2 + E^4*(3750 + 4687500*x)*Log[x^(-1)] + 390625*E^8*Log[x^(-1)]^2),x]

[Out]

x + 1171875*E^4*Defer[Int][(3 + 3750*x + 625*E^4*Log[x^(-1)])^(-2), x] - 7031250*Defer[Int][x/(3 + 3750*x + 62
5*E^4*Log[x^(-1)])^2, x] + 1875*Defer[Int][(3 + 3750*x + 625*E^4*Log[x^(-1)])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5634 \left (1+\frac {390625 e^4}{1878}\right )+22500 x+14062500 x^2+e^4 (1175625+4687500 x) \log \left (\frac {1}{x}\right )+390625 e^8 \log ^2\left (\frac {1}{x}\right )}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2} \, dx\\ &=\int \left (1+\frac {1171875 \left (e^4-6 x\right )}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2}+\frac {1875}{3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )}\right ) \, dx\\ &=x+1875 \int \frac {1}{3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )} \, dx+1171875 \int \frac {e^4-6 x}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2} \, dx\\ &=x+1875 \int \frac {1}{3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )} \, dx+1171875 \int \left (\frac {e^4}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2}-\frac {6 x}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2}\right ) \, dx\\ &=x+1875 \int \frac {1}{3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )} \, dx-7031250 \int \frac {x}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2} \, dx+\left (1171875 e^4\right ) \int \frac {1}{\left (3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 21, normalized size = 0.84 \begin {gather*} x+\frac {1875 x}{3+3750 x+625 e^4 \log \left (\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5634 + 1171875*E^4 + 22500*x + 14062500*x^2 + E^4*(1175625 + 4687500*x)*Log[x^(-1)] + 390625*E^8*Lo
g[x^(-1)]^2)/(9 + 22500*x + 14062500*x^2 + E^4*(3750 + 4687500*x)*Log[x^(-1)] + 390625*E^8*Log[x^(-1)]^2),x]

[Out]

x + (1875*x)/(3 + 3750*x + 625*E^4*Log[x^(-1)])

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fricas [A]  time = 0.66, size = 34, normalized size = 1.36 \begin {gather*} \frac {625 \, x e^{4} \log \left (\frac {1}{x}\right ) + 3750 \, x^{2} + 1878 \, x}{625 \, e^{4} \log \left (\frac {1}{x}\right ) + 3750 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((390625*exp(4)^2*log(1/x)^2+(4687500*x+1175625)*exp(4)*log(1/x)+1171875*exp(4)[phone]*x^2+22500*x+
5634)/(390625*exp(4)^2*log(1/x)^2+(4687500*x+3750)*exp(4)*log(1/x)[phone]*x^2+22500*x+9),x, algorithm="frica
s")

[Out]

(625*x*e^4*log(1/x) + 3750*x^2 + 1878*x)/(625*e^4*log(1/x) + 3750*x + 3)

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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((390625*exp(4)^2*log(1/x)^2+(4687500*x+1175625)*exp(4)*log(1/x)+1171875*exp(4)[phone]*x^2+22500*x+
5634)/(390625*exp(4)^2*log(1/x)^2+(4687500*x+3750)*exp(4)*log(1/x)[phone]*x^2+22500*x+9),x, algorithm="giac"
)

[Out]

Timed out

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maple [A]  time = 0.30, size = 21, normalized size = 0.84

method	result	size
risch	\(x +\frac {1875 x}{625 \,{\mathrm e}^{4} \ln \left (\frac {1}{x}\right )+3750 x +3}\)	\(21\)
norman	\(\frac {1878 x +3750 x^{2}+625 \,{\mathrm e}^{4} \ln \left (\frac {1}{x}\right ) x}{625 \,{\mathrm e}^{4} \ln \left (\frac {1}{x}\right )+3750 x +3}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((390625*exp(4)^2*ln(1/x)^2+(4687500*x+1175625)*exp(4)*ln(1/x)+1171875*exp(4)[phone]*x^2+22500*x+5634)/(3
90625*exp(4)^2*ln(1/x)^2+(4687500*x+3750)*exp(4)*ln(1/x)[phone]*x^2+22500*x+9),x,method=_RETURNVERBOSE)

[Out]

x+1875*x/(625*exp(4)*ln(1/x)+3750*x+3)

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maxima [A]  time = 0.38, size = 30, normalized size = 1.20 \begin {gather*} \frac {625 \, x e^{4} \log \relax (x) - 3750 \, x^{2} - 1878 \, x}{625 \, e^{4} \log \relax (x) - 3750 \, x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((390625*exp(4)^2*log(1/x)^2+(4687500*x+1175625)*exp(4)*log(1/x)+1171875*exp(4)[phone]*x^2+22500*x+
5634)/(390625*exp(4)^2*log(1/x)^2+(4687500*x+3750)*exp(4)*log(1/x)[phone]*x^2+22500*x+9),x, algorithm="maxim
a")

[Out]

(625*x*e^4*log(x) - 3750*x^2 - 1878*x)/(625*e^4*log(x) - 3750*x - 3)

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mupad [B]  time = 5.27, size = 20, normalized size = 0.80 \begin {gather*} x+\frac {1875\,x}{3750\,x+625\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^4+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22500*x + 1171875*exp(4) + 390625*log(1/x)^2*exp(8) + 14062500*x^2 + log(1/x)*exp(4)*(4687500*x + 1175625
) + 5634)/(22500*x + 390625*log(1/x)^2*exp(8) + 14062500*x^2 + log(1/x)*exp(4)*(4687500*x + 3750) + 9),x)

[Out]

x + (1875*x)/(3750*x + 625*log(1/x)*exp(4) + 3)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.76 \begin {gather*} x + \frac {1875 x}{3750 x + 625 e^{4} \log {\left (\frac {1}{x} \right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((390625*exp(4)**2*ln(1/x)**2+(4687500*x+1175625)*exp(4)*ln(1/x)+1171875*exp(4)[phone]*x**2+22500*x
+5634)/(390625*exp(4)**2*ln(1/x)**2+(4687500*x+3750)*exp(4)*ln(1/x)[phone]*x**2+22500*x+9),x)

[Out]

x + 1875*x/(3750*x + 625*exp(4)*log(1/x) + 3)

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