3.44.62 \(\int \frac {-1-e^4+6 x-3 x^2+e^{2 x} (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3)}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} (-7+e^4 (-7+x)+x-7 x^2+x^3)} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\frac {1}{\left (3+e^{2 x} (7-x)-x\right ) \left (1+e^4+x^2\right )}\right ) \]

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Rubi [F]  time = 1.22, 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 {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 - E^4 + 6*x - 3*x^2 + E^(2*x)*(13 + E^4*(13 - 2*x) + 12*x + 11*x^2 - 2*x^3))/(-3 + E^4*(-3 + x) + x -
3*x^2 + x^3 + E^(2*x)*(-7 + E^4*(-7 + x) + x - 7*x^2 + x^3)),x]

[Out]

-2*x - Log[7 - x] - Log[1 + E^4 + x^2] - 6*Defer[Int][(-3 + E^(2*x)*(-7 + x) + x)^(-1), x] + 2*Defer[Int][x/(-
3 + E^(2*x)*(-7 + x) + x), x] + 4*Defer[Int][1/((-7 + x)*(-3 - 7*E^(2*x) + x + E^(2*x)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+e^4-6 x+3 x^2-e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{\left (3+7 e^{2 x}-x-e^{2 x} x\right ) \left (1+e^4+x^2\right )} \, dx\\ &=\int \left (\frac {2 \left (23-10 x+x^2\right )}{(-7+x) \left (-3-7 e^{2 x}+x+e^{2 x} x\right )}+\frac {-13 \left (1+e^4\right )-2 \left (6-e^4\right ) x-11 x^2+2 x^3}{(7-x) \left (1+e^4+x^2\right )}\right ) \, dx\\ &=2 \int \frac {23-10 x+x^2}{(-7+x) \left (-3-7 e^{2 x}+x+e^{2 x} x\right )} \, dx+\int \frac {-13 \left (1+e^4\right )-2 \left (6-e^4\right ) x-11 x^2+2 x^3}{(7-x) \left (1+e^4+x^2\right )} \, dx\\ &=2 \int \left (-\frac {3}{-3-7 e^{2 x}+x+e^{2 x} x}+\frac {2}{(-7+x) \left (-3-7 e^{2 x}+x+e^{2 x} x\right )}+\frac {x}{-3-7 e^{2 x}+x+e^{2 x} x}\right ) \, dx+\int \left (-2+\frac {1}{7-x}-\frac {2 x}{1+e^4+x^2}\right ) \, dx\\ &=-2 x-\log (7-x)+2 \int \frac {x}{-3-7 e^{2 x}+x+e^{2 x} x} \, dx-2 \int \frac {x}{1+e^4+x^2} \, dx+4 \int \frac {1}{(-7+x) \left (-3-7 e^{2 x}+x+e^{2 x} x\right )} \, dx-6 \int \frac {1}{-3-7 e^{2 x}+x+e^{2 x} x} \, dx\\ &=-2 x-\log (7-x)-\log \left (1+e^4+x^2\right )+2 \int \frac {x}{-3+e^{2 x} (-7+x)+x} \, dx+4 \int \frac {1}{(-7+x) \left (-3-7 e^{2 x}+x+e^{2 x} x\right )} \, dx-6 \int \frac {1}{-3+e^{2 x} (-7+x)+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 35, normalized size = 1.17 \begin {gather*} -\log \left (3+7 e^{2 x}-x-e^{2 x} x\right )-\log \left (1+e^4+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - E^4 + 6*x - 3*x^2 + E^(2*x)*(13 + E^4*(13 - 2*x) + 12*x + 11*x^2 - 2*x^3))/(-3 + E^4*(-3 + x)
+ x - 3*x^2 + x^3 + E^(2*x)*(-7 + E^4*(-7 + x) + x - 7*x^2 + x^3)),x]

[Out]

-Log[3 + 7*E^(2*x) - x - E^(2*x)*x] - Log[1 + E^4 + x^2]

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fricas [A]  time = 0.52, size = 41, normalized size = 1.37 \begin {gather*} -\log \left (x^{3} - 7 \, x^{2} + {\left (x - 7\right )} e^{4} + x - 7\right ) - \log \left (\frac {{\left (x - 7\right )} e^{\left (2 \, x\right )} + x - 3}{x - 7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/(((x-7)*exp(4)+x^3-7*x^2+x-7)*
exp(x)^2+(x-3)*exp(4)+x^3-3*x^2+x-3),x, algorithm="fricas")

[Out]

-log(x^3 - 7*x^2 + (x - 7)*e^4 + x - 7) - log(((x - 7)*e^(2*x) + x - 3)/(x - 7))

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giac [A]  time = 0.28, size = 29, normalized size = 0.97 \begin {gather*} -\log \left (x^{2} + e^{4} + 1\right ) - \log \left (x e^{\left (2 \, x\right )} + x - 7 \, e^{\left (2 \, x\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/(((x-7)*exp(4)+x^3-7*x^2+x-7)*
exp(x)^2+(x-3)*exp(4)+x^3-3*x^2+x-3),x, algorithm="giac")

[Out]

-log(x^2 + e^4 + 1) - log(x*e^(2*x) + x - 7*e^(2*x) - 3)

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maple [A]  time = 0.20, size = 30, normalized size = 1.00

method	result	size
norman	\(-\ln \left (x^{2}+{\mathrm e}^{4}+1\right )-\ln \left (x \,{\mathrm e}^{2 x}-7 \,{\mathrm e}^{2 x}+x -3\right )\)	\(30\)
risch	\(-\ln \left (x^{3}-7 x^{2}+\left ({\mathrm e}^{4}+1\right ) x -7 \,{\mathrm e}^{4}-7\right )-\ln \left ({\mathrm e}^{2 x}+\frac {x -3}{x -7}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/(((x-7)*exp(4)+x^3-7*x^2+x-7)*exp(x)
^2+(x-3)*exp(4)+x^3-3*x^2+x-3),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+exp(4)+1)-ln(x*exp(x)^2-7*exp(x)^2+x-3)

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maxima [A]  time = 0.40, size = 37, normalized size = 1.23 \begin {gather*} -\log \left (x^{2} + e^{4} + 1\right ) - \log \left (x - 7\right ) - \log \left (\frac {{\left (x - 7\right )} e^{\left (2 \, x\right )} + x - 3}{x - 7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/(((x-7)*exp(4)+x^3-7*x^2+x-7)*
exp(x)^2+(x-3)*exp(4)+x^3-3*x^2+x-3),x, algorithm="maxima")

[Out]

-log(x^2 + e^4 + 1) - log(x - 7) - log(((x - 7)*e^(2*x) + x - 3)/(x - 7))

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mupad [B]  time = 3.40, size = 29, normalized size = 0.97 \begin {gather*} -\ln \left (x^2+{\mathrm {e}}^4+1\right )-\ln \left (x-7\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4) - 6*x - exp(2*x)*(12*x + 11*x^2 - 2*x^3 - exp(4)*(2*x - 13) + 13) + 3*x^2 + 1)/(x + exp(2*x)*(x +
exp(4)*(x - 7) - 7*x^2 + x^3 - 7) + exp(4)*(x - 3) - 3*x^2 + x^3 - 3),x)

[Out]

- log(exp(4) + x^2 + 1) - log(x - 7*exp(2*x) + x*exp(2*x) - 3)

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sympy [A]  time = 0.64, size = 37, normalized size = 1.23 \begin {gather*} - \log {\left (e^{2 x} + \frac {x - 3}{x - 7} \right )} - \log {\left (x^{3} - 7 x^{2} + x \left (1 + e^{4}\right ) - 7 e^{4} - 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x+13)*exp(4)-2*x**3+11*x**2+12*x+13)*exp(x)**2-exp(4)-3*x**2+6*x-1)/(((x-7)*exp(4)+x**3-7*x**2
+x-7)*exp(x)**2+(x-3)*exp(4)+x**3-3*x**2+x-3),x)

[Out]

-log(exp(2*x) + (x - 3)/(x - 7)) - log(x**3 - 7*x**2 + x*(1 + exp(4)) - 7*exp(4) - 7)

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