3.72.57 \(\int \frac {-45-20 x-10 x^2+e^{2 x} (-2 x-x^2)+e^x (-9-5 x-x^2-3 x^3)}{45 x+55 x^2+10 x^3+e^{2 x} (x^2+x^3)+e^x (9 x+16 x^2+7 x^3)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\frac {1+x}{2 x \left (\frac {3 (3-x)}{5+e^x}+x\right )}\right ) \]

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Rubi [F]  time = 2.06, 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 {-45-20 x-10 x^2+e^{2 x} \left (-2 x-x^2\right )+e^x \left (-9-5 x-x^2-3 x^3\right )}{45 x+55 x^2+10 x^3+e^{2 x} \left (x^2+x^3\right )+e^x \left (9 x+16 x^2+7 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-45 - 20*x - 10*x^2 + E^(2*x)*(-2*x - x^2) + E^x*(-9 - 5*x - x^2 - 3*x^3))/(45*x + 55*x^2 + 10*x^3 + E^(2
*x)*(x^2 + x^3) + E^x*(9*x + 16*x^2 + 7*x^3)),x]

[Out]

-x + Log[5 + E^x] - 2*Log[x] + Log[1 + x] + 9*Defer[Int][(9 + 2*x + E^x*x)^(-1), x] + 9*Defer[Int][1/(x*(9 + 2
*x + E^x*x)), x] + 2*Defer[Int][x/(9 + 2*x + E^x*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45-20 x-10 x^2+e^{2 x} \left (-2 x-x^2\right )+e^x \left (-9-5 x-x^2-3 x^3\right )}{\left (5+e^x\right ) x (1+x) \left (9+2 x+e^x x\right )} \, dx\\ &=\int \left (-\frac {5}{5+e^x}+\frac {-2-x}{x (1+x)}+\frac {9+9 x+2 x^2}{x \left (9+2 x+e^x x\right )}\right ) \, dx\\ &=-\left (5 \int \frac {1}{5+e^x} \, dx\right )+\int \frac {-2-x}{x (1+x)} \, dx+\int \frac {9+9 x+2 x^2}{x \left (9+2 x+e^x x\right )} \, dx\\ &=-\left (5 \operatorname {Subst}\left (\int \frac {1}{x (5+x)} \, dx,x,e^x\right )\right )+\int \left (-\frac {2}{x}+\frac {1}{1+x}\right ) \, dx+\int \left (\frac {9}{9+2 x+e^x x}+\frac {9}{x \left (9+2 x+e^x x\right )}+\frac {2 x}{9+2 x+e^x x}\right ) \, dx\\ &=-2 \log (x)+\log (1+x)+2 \int \frac {x}{9+2 x+e^x x} \, dx+9 \int \frac {1}{9+2 x+e^x x} \, dx+9 \int \frac {1}{x \left (9+2 x+e^x x\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {1}{5+x} \, dx,x,e^x\right )\\ &=-x+\log \left (5+e^x\right )-2 \log (x)+\log (1+x)+2 \int \frac {x}{9+2 x+e^x x} \, dx+9 \int \frac {1}{9+2 x+e^x x} \, dx+9 \int \frac {1}{x \left (9+2 x+e^x x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 28, normalized size = 0.97 \begin {gather*} \log \left (5+e^x\right )-\log (x)+\log (1+x)-\log \left (9+2 x+e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-45 - 20*x - 10*x^2 + E^(2*x)*(-2*x - x^2) + E^x*(-9 - 5*x - x^2 - 3*x^3))/(45*x + 55*x^2 + 10*x^3
+ E^(2*x)*(x^2 + x^3) + E^x*(9*x + 16*x^2 + 7*x^3)),x]

[Out]

Log[5 + E^x] - Log[x] + Log[1 + x] - Log[9 + 2*x + E^x*x]

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fricas [A]  time = 0.78, size = 30, normalized size = 1.03 \begin {gather*} \log \left (x + 1\right ) - 2 \, \log \relax (x) - \log \left (\frac {x e^{x} + 2 \, x + 9}{x}\right ) + \log \left (e^{x} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-2*x)*exp(x)^2+(-3*x^3-x^2-5*x-9)*exp(x)-10*x^2-20*x-45)/((x^3+x^2)*exp(x)^2+(7*x^3+16*x^2+9*x
)*exp(x)+10*x^3+55*x^2+45*x),x, algorithm="fricas")

[Out]

log(x + 1) - 2*log(x) - log((x*e^x + 2*x + 9)/x) + log(e^x + 5)

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giac [A]  time = 1.59, size = 26, normalized size = 0.90 \begin {gather*} -\log \left (x e^{x} + 2 \, x + 9\right ) + \log \left (x + 1\right ) - \log \relax (x) + \log \left (e^{x} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-2*x)*exp(x)^2+(-3*x^3-x^2-5*x-9)*exp(x)-10*x^2-20*x-45)/((x^3+x^2)*exp(x)^2+(7*x^3+16*x^2+9*x
)*exp(x)+10*x^3+55*x^2+45*x),x, algorithm="giac")

[Out]

-log(x*e^x + 2*x + 9) + log(x + 1) - log(x) + log(e^x + 5)

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maple [A]  time = 0.06, size = 27, normalized size = 0.93

method	result	size
norman	\(-\ln \relax (x )-\ln \left ({\mathrm e}^{x} x +2 x +9\right )+\ln \left (x +1\right )+\ln \left ({\mathrm e}^{x}+5\right )\)	\(27\)
risch	\(-2 \ln \relax (x )+\ln \left (x +1\right )+\ln \left ({\mathrm e}^{x}+5\right )-\ln \left ({\mathrm e}^{x}+\frac {2 x +9}{x}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-2*x)*exp(x)^2+(-3*x^3-x^2-5*x-9)*exp(x)-10*x^2-20*x-45)/((x^3+x^2)*exp(x)^2+(7*x^3+16*x^2+9*x)*exp(
x)+10*x^3+55*x^2+45*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-ln(exp(x)*x+2*x+9)+ln(x+1)+ln(exp(x)+5)

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maxima [A]  time = 0.47, size = 30, normalized size = 1.03 \begin {gather*} \log \left (x + 1\right ) - 2 \, \log \relax (x) - \log \left (\frac {x e^{x} + 2 \, x + 9}{x}\right ) + \log \left (e^{x} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-2*x)*exp(x)^2+(-3*x^3-x^2-5*x-9)*exp(x)-10*x^2-20*x-45)/((x^3+x^2)*exp(x)^2+(7*x^3+16*x^2+9*x
)*exp(x)+10*x^3+55*x^2+45*x),x, algorithm="maxima")

[Out]

log(x + 1) - 2*log(x) - log((x*e^x + 2*x + 9)/x) + log(e^x + 5)

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mupad [B]  time = 0.14, size = 26, normalized size = 0.90 \begin {gather*} \ln \left (x+1\right )-\ln \left (2\,x+x\,{\mathrm {e}}^x+9\right )+\ln \left ({\mathrm {e}}^x+5\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x + exp(x)*(5*x + x^2 + 3*x^3 + 9) + exp(2*x)*(2*x + x^2) + 10*x^2 + 45)/(45*x + exp(2*x)*(x^2 + x^3)
+ 55*x^2 + 10*x^3 + exp(x)*(9*x + 16*x^2 + 7*x^3)),x)

[Out]

log(x + 1) - log(2*x + x*exp(x) + 9) + log(exp(x) + 5) - log(x)

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sympy [A]  time = 0.51, size = 29, normalized size = 1.00 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x + 1 \right )} + \log {\left (e^{x} + 5 \right )} - \log {\left (e^{x} + \frac {4 x + 18}{2 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-2*x)*exp(x)**2+(-3*x**3-x**2-5*x-9)*exp(x)-10*x**2-20*x-45)/((x**3+x**2)*exp(x)**2+(7*x**3+1
6*x**2+9*x)*exp(x)+10*x**3+55*x**2+45*x),x)

[Out]

-2*log(x) + log(x + 1) + log(exp(x) + 5) - log(exp(x) + (4*x + 18)/(2*x))

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