3.48.10 \(\int \frac {e^{11}+e^{11+x} (-1-x)+4 x^3}{e^{11} (-7-x)+e^{11+x} x-x^4} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {2}{7+x-e^x x+\frac {x^4}{e^{11}}}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6684} \begin {gather*} -\log \left (x^4-e^{x+11} x+e^{11} (x+7)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[-(E^(11 + x)*x) + x^4 + E^11*(7 + x)]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (-e^{11+x} x+x^4+e^{11} (7+x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 27, normalized size = 1.29 \begin {gather*} -\log \left (-7 e^{11}-e^{11} x+e^{11+x} x-x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[-7*E^11 - E^11*x + E^(11 + x)*x - x^4]

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fricas [A]  time = 0.57, size = 30, normalized size = 1.43 \begin {gather*} -\log \relax (x) - \log \left (-\frac {x^{4} + {\left (x + 7\right )} e^{11} - x e^{\left (x + 11\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) - log(-(x^4 + (x + 7)*e^11 - x*e^(x + 11))/x)

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giac [A]  time = 0.36, size = 24, normalized size = 1.14 \begin {gather*} -\log \left (-x^{4} - x e^{11} + x e^{\left (x + 11\right )} - 7 \, e^{11}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x^4 - x*e^11 + x*e^(x + 11) - 7*e^11)

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maple [A]  time = 0.09, size = 25, normalized size = 1.19

method	result	size
norman	\(-\ln \left (-x^{4}+x \,{\mathrm e}^{11} {\mathrm e}^{x}-{\mathrm e}^{11} x -7 \,{\mathrm e}^{11}\right )\)	\(25\)
risch	\(-\ln \relax (x )-\ln \left ({\mathrm e}^{x}-\frac {\left (x^{4}+{\mathrm e}^{11} x +7 \,{\mathrm e}^{11}\right ) {\mathrm e}^{-11}}{x}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-x^4+x*exp(11)*exp(x)-exp(11)*x-7*exp(11))

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maxima [A]  time = 0.40, size = 34, normalized size = 1.62 \begin {gather*} -\log \relax (x) - \log \left (-\frac {{\left (x^{4} + x e^{11} - x e^{\left (x + 11\right )} + 7 \, e^{11}\right )} e^{\left (-11\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) - log(-(x^4 + x*e^11 - x*e^(x + 11) + 7*e^11)*e^(-11)/x)

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mupad [B]  time = 3.27, size = 22, normalized size = 1.05 \begin {gather*} -\ln \left (7\,{\mathrm {e}}^{11}-x\,{\mathrm {e}}^{x+11}+x\,{\mathrm {e}}^{11}+x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(7*exp(11) - x*exp(x + 11) + x*exp(11) + x^4)

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sympy [A]  time = 0.22, size = 29, normalized size = 1.38 \begin {gather*} - \log {\relax (x )} - \log {\left (e^{x} + \frac {- x^{4} - x e^{11} - 7 e^{11}}{x e^{11}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*exp(11)*exp(x)+exp(11)+4*x**3)/(x*exp(11)*exp(x)+(-x-7)*exp(11)-x**4),x)

[Out]

-log(x) - log(exp(x) + (-x**4 - x*exp(11) - 7*exp(11))*exp(-11)/x)

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