3.38.52 \(\int \frac {-2+e^{e^x+x}+40 e^{20 x^2} x}{-5+e^5+e^{e^x}+e^{20 x^2}-2 x} \, dx\)

Optimal. Leaf size=27 \[ \log \left (5-e^5-e^{e^x}-e^{20 x^2}+2 x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6684} \begin {gather*} \log \left (-e^{20 x^2}+2 x-e^{e^x}-e^5+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^(E^x + x) + 40*E^(20*x^2)*x)/(-5 + E^5 + E^E^x + E^(20*x^2) - 2*x),x]

[Out]

Log[5 - E^5 - E^E^x - E^(20*x^2) + 2*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 (5-e^5-e^{e^x}-e^{20 x^2}+2 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 21, normalized size = 0.78 \begin {gather*} \log \left (-5+e^5+e^{e^x}+e^{20 x^2}-2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^(E^x + x) + 40*E^(20*x^2)*x)/(-5 + E^5 + E^E^x + E^(20*x^2) - 2*x),x]

[Out]

Log[-5 + E^5 + E^E^x + E^(20*x^2) - 2*x]

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fricas [A]  time = 0.99, size = 32, normalized size = 1.19 \begin {gather*} -x + \log \left (-{\left (2 \, x - e^{5} + 5\right )} e^{x} + e^{\left (20 \, x^{2} + x\right )} + e^{\left (x + e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*exp(exp(x))+40*x*exp(20*x^2)-2)/(exp(exp(x))+exp(20*x^2)+exp(5)-2*x-5),x, algorithm="fricas"
)

[Out]

-x + log(-(2*x - e^5 + 5)*e^x + e^(20*x^2 + x) + e^(x + e^x))

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giac [A]  time = 0.17, size = 38, normalized size = 1.41 \begin {gather*} -x + \log \left (2 \, x e^{x} - e^{\left (20 \, x^{2} + x\right )} - e^{\left (x + e^{x}\right )} - e^{\left (x + 5\right )} + 5 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*exp(exp(x))+40*x*exp(20*x^2)-2)/(exp(exp(x))+exp(20*x^2)+exp(5)-2*x-5),x, algorithm="giac")

[Out]

-x + log(2*x*e^x - e^(20*x^2 + x) - e^(x + e^x) - e^(x + 5) + 5*e^x)

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maple [A]  time = 0.08, size = 18, normalized size = 0.67

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{20 x^{2}}+{\mathrm e}^{5}-2 x -5\right )\)	\(18\)
default	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{20 x^{2}}+{\mathrm e}^{5}-2 x -5\right )\)	\(18\)
norman	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{20 x^{2}}+{\mathrm e}^{5}-2 x -5\right )\)	\(18\)
risch	\(\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{20 x^{2}}+{\mathrm e}^{5}-2 x -5\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*exp(exp(x))+40*x*exp(20*x^2)-2)/(exp(exp(x))+exp(20*x^2)+exp(5)-2*x-5),x,method=_RETURNVERBOSE)

[Out]

ln(exp(exp(x))+exp(20*x^2)+exp(5)-2*x-5)

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maxima [A]  time = 0.35, size = 23, normalized size = 0.85 \begin {gather*} \log \left (2 \, x - e^{5} - e^{\left (20 \, x^{2}\right )} - e^{\left (e^{x}\right )} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*exp(exp(x))+40*x*exp(20*x^2)-2)/(exp(exp(x))+exp(20*x^2)+exp(5)-2*x-5),x, algorithm="maxima"
)

[Out]

log(2*x - e^5 - e^(20*x^2) - e^(e^x) + 5)

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mupad [B]  time = 0.23, size = 23, normalized size = 0.85 \begin {gather*} \ln \left (2\,x-{\mathrm {e}}^{{\mathrm {e}}^x}-{\mathrm {e}}^5-{\mathrm {e}}^{20\,x^2}+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x*exp(20*x^2) + exp(exp(x))*exp(x) - 2)/(exp(exp(x)) - 2*x + exp(5) + exp(20*x^2) - 5),x)

[Out]

log(2*x - exp(exp(x)) - exp(5) - exp(20*x^2) + 5)

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sympy [A]  time = 0.20, size = 20, normalized size = 0.74 \begin {gather*} \log {\left (- 2 x + e^{20 x^{2}} + e^{e^{x}} - 5 + e^{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*exp(exp(x))+40*x*exp(20*x**2)-2)/(exp(exp(x))+exp(20*x**2)+exp(5)-2*x-5),x)

[Out]

log(-2*x + exp(20*x**2) + exp(exp(x)) - 5 + exp(5))

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