3.73.34 \(\int \frac {27+3 e^{4 x}+18 x+3 x^2+e (-3 x-82 x^2)+e^{2 x} (18+6 x+e (-6 x-164 x^2))}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} (3 x+82 x^2)+e^{2 x} (18 x+498 x^2+164 x^3)} \, dx\)

Optimal. Leaf size=24 \[ \frac {e}{3+e^{2 x}+x}+\log \left (\frac {3 x}{3+82 x}\right ) \]

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Rubi [F]  time = 1.21, 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 {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x + 756*
x^2 + 495*x^3 + 82*x^4 + E^(4*x)*(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]

[Out]

Log[x] - Log[3 + 82*x] + 5*E*Defer[Int][(3 + E^(2*x) + x)^(-2), x] + 2*E*Defer[Int][x/(3 + E^(2*x) + x)^2, x]
- 2*E*Defer[Int][(3 + E^(2*x) + x)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{4 x}+6 e^{2 x} (3+x)+3 (3+x)^2-e x (3+82 x)-2 e^{1+2 x} x (3+82 x)}{x \left (3+e^{2 x}+x\right )^2 (3+82 x)} \, dx\\ &=\int \left (-\frac {2 e}{3+e^{2 x}+x}+\frac {e (5+2 x)}{\left (3+e^{2 x}+x\right )^2}+\frac {3}{x (3+82 x)}\right ) \, dx\\ &=3 \int \frac {1}{x (3+82 x)} \, dx+e \int \frac {5+2 x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx\\ &=-\left (82 \int \frac {1}{3+82 x} \, dx\right )+e \int \left (\frac {5}{\left (3+e^{2 x}+x\right )^2}+\frac {2 x}{\left (3+e^{2 x}+x\right )^2}\right ) \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+\int \frac {1}{x} \, dx\\ &=\log (x)-\log (3+82 x)+(2 e) \int \frac {x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+(5 e) \int \frac {1}{\left (3+e^{2 x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 23, normalized size = 0.96 \begin {gather*} \frac {e}{3+e^{2 x}+x}+\log (x)-\log (3+82 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x
+ 756*x^2 + 495*x^3 + 82*x^4 + E^(4*x)*(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]

[Out]

E/(3 + E^(2*x) + x) + Log[x] - Log[3 + 82*x]

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fricas [A]  time = 0.70, size = 41, normalized size = 1.71 \begin {gather*} -\frac {{\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (82 \, x + 3\right ) - {\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \relax (x) - e}{x + e^{\left (2 \, x\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="fricas")

[Out]

-((x + e^(2*x) + 3)*log(82*x + 3) - (x + e^(2*x) + 3)*log(x) - e)/(x + e^(2*x) + 3)

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giac [B]  time = 0.23, size = 60, normalized size = 2.50 \begin {gather*} -\frac {x \log \left (82 \, x + 3\right ) + e^{\left (2 \, x\right )} \log \left (82 \, x + 3\right ) - x \log \relax (x) - e^{\left (2 \, x\right )} \log \relax (x) - e + 3 \, \log \left (82 \, x + 3\right ) - 3 \, \log \relax (x)}{x + e^{\left (2 \, x\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="giac")

[Out]

-(x*log(82*x + 3) + e^(2*x)*log(82*x + 3) - x*log(x) - e^(2*x)*log(x) - e + 3*log(82*x + 3) - 3*log(x))/(x + e
^(2*x) + 3)

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maple [A]  time = 0.15, size = 24, normalized size = 1.00

method	result	size
norman	\(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \relax (x )\)	\(24\)
risch	\(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(
x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x,method=_RETURNVERBOSE)

[Out]

exp(1)/(exp(x)^2+3+x)-ln(82*x+3)+ln(x)

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maxima [A]  time = 0.41, size = 23, normalized size = 0.96 \begin {gather*} \frac {e}{x + e^{\left (2 \, x\right )} + 3} - \log \left (82 \, x + 3\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x
)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="maxima")

[Out]

e/(x + e^(2*x) + 3) - log(82*x + 3) + log(x)

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mupad [B]  time = 4.43, size = 34, normalized size = 1.42 \begin {gather*} \ln \relax (x)-\ln \left (x+\frac {3}{82}\right )-\frac {\frac {{\mathrm {e}}^{2\,x+1}}{3}+\frac {x\,\mathrm {e}}{3}}{x+{\mathrm {e}}^{2\,x}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + 3*exp(4*x) + exp(2*x)*(6*x - exp(1)*(6*x + 164*x^2) + 18) - exp(1)*(3*x + 82*x^2) + 3*x^2 + 27)/(2
7*x + exp(4*x)*(3*x + 82*x^2) + exp(2*x)*(18*x + 498*x^2 + 164*x^3) + 756*x^2 + 495*x^3 + 82*x^4),x)

[Out]

log(x) - log(x + 3/82) - (exp(2*x + 1)/3 + (x*exp(1))/3)/(x + exp(2*x) + 3)

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sympy [A]  time = 0.21, size = 20, normalized size = 0.83 \begin {gather*} \log {\relax (x )} - \log {\left (x + \frac {3}{82} \right )} + \frac {e}{x + e^{2 x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)**4+((-164*x**2-6*x)*exp(1)+18+6*x)*exp(x)**2+(-82*x**2-3*x)*exp(1)+3*x**2+18*x+27)/((82*x*
*2+3*x)*exp(x)**4+(164*x**3+498*x**2+18*x)*exp(x)**2+82*x**4+495*x**3+756*x**2+27*x),x)

[Out]

log(x) - log(x + 3/82) + E/(x + exp(2*x) + 3)

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