3.21.46 \(\int \frac {1+e^x (1+x)+\log (i \pi +\log (-2+e^8))}{x+e^x x+x \log (i \pi +\log (-2+e^8))} \, dx\)

Optimal. Leaf size=21 \[ \log \left (x \left (1+e^x+\log \left (i \pi +\log \left (-2+e^8\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6, 6684} \begin {gather*} \log \left (e^x x+x \left (1+\log \left (\log \left (e^8-2\right )+i \pi \right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^x*(1 + x) + Log[I*Pi + Log[-2 + E^8]])/(x + E^x*x + x*Log[I*Pi + Log[-2 + E^8]]),x]

[Out]

Log[E^x*x + x*(1 + Log[I*Pi + Log[-2 + E^8]])]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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} &=\int \frac {1+e^x (1+x)+\log \left (i \pi +\log \left (-2+e^8\right )\right )}{e^x x+x \left (1+\log \left (i \pi +\log \left (-2+e^8\right )\right )\right )} \, dx\\ &=\log \left (e^x x+x \left (1+\log \left (i \pi +\log \left (-2+e^8\right )\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 1.05 \begin {gather*} \log (x)+\log \left (1+e^x+\log \left (i \pi +\log \left (-2+e^8\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x*(1 + x) + Log[I*Pi + Log[-2 + E^8]])/(x + E^x*x + x*Log[I*Pi + Log[-2 + E^8]]),x]

[Out]

Log[x] + Log[1 + E^x + Log[I*Pi + Log[-2 + E^8]]]

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fricas [A]  time = 0.77, size = 16, normalized size = 0.76 \begin {gather*} \log \relax (x) + \log \left (e^{x} + \log \left (\log \left (-e^{8} + 2\right )\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(-exp(4)^2+2))+(x+1)*exp(x)+1)/(x*log(log(-exp(4)^2+2))+exp(x)*x+x),x, algorithm="fricas")

[Out]

log(x) + log(e^x + log(log(-e^8 + 2)) + 1)

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giac [A]  time = 0.23, size = 16, normalized size = 0.76 \begin {gather*} \log \relax (x) + \log \left (e^{x} + \log \left (\log \left (-e^{8} + 2\right )\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(-exp(4)^2+2))+(x+1)*exp(x)+1)/(x*log(log(-exp(4)^2+2))+exp(x)*x+x),x, algorithm="giac")

[Out]

log(x) + log(e^x + log(log(-e^8 + 2)) + 1)

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maple [A]  time = 0.14, size = 17, normalized size = 0.81

method	result	size
risch	\(\ln \relax (x )+\ln \left (\ln \left (\ln \left (-{\mathrm e}^{8}+2\right )\right )+{\mathrm e}^{x}+1\right )\)	\(17\)
norman	\(\ln \relax (x )+\ln \left (\ln \left (\ln \left (-{\mathrm e}^{8}+2\right )\right )+{\mathrm e}^{x}+1\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(ln(-exp(4)^2+2))+(x+1)*exp(x)+1)/(x*ln(ln(-exp(4)^2+2))+exp(x)*x+x),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(ln(-exp(8)+2))+exp(x)+1)

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maxima [A]  time = 0.99, size = 16, normalized size = 0.76 \begin {gather*} \log \relax (x) + \log \left (e^{x} + \log \left (\log \left (-e^{8} + 2\right )\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(log(-exp(4)^2+2))+(x+1)*exp(x)+1)/(x*log(log(-exp(4)^2+2))+exp(x)*x+x),x, algorithm="maxima")

[Out]

log(x) + log(e^x + log(log(-e^8 + 2)) + 1)

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mupad [B]  time = 0.09, size = 16, normalized size = 0.76 \begin {gather*} \ln \left ({\mathrm {e}}^x+\ln \left (\ln \left (2-{\mathrm {e}}^8\right )\right )+1\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2 - exp(8))) + exp(x)*(x + 1) + 1)/(x + x*exp(x) + x*log(log(2 - exp(8)))),x)

[Out]

log(exp(x) + log(log(2 - exp(8))) + 1) + log(x)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.95 \begin {gather*} \log {\relax (x )} + \log {\left (e^{x} + 1 + \log {\left (\log {\left (-2 + e^{8} \right )} + i \pi \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(ln(-exp(4)**2+2))+(x+1)*exp(x)+1)/(x*ln(ln(-exp(4)**2+2))+exp(x)*x+x),x)

[Out]

log(x) + log(exp(x) + 1 + log(log(-2 + exp(8)) + I*pi))

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