3.93.10 \(\int \frac {3+e^4+x-\log (5)}{3+e^4+x} \, dx\)

Optimal. Leaf size=24 \[ x-\log (5) \left (\log ^2(3)+\log \left (1+\frac {1}{4} \left (-1+e^4+x\right )\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {43} \begin {gather*} x-\log (5) \log \left (x+e^4+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E^4 + x - Log[5])/(3 + E^4 + x),x]

[Out]

x - Log[5]*Log[3 + E^4 + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {\log (5)}{3+e^4+x}\right ) \, dx\\ &=x-\log (5) \log \left (3+e^4+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.54 \begin {gather*} x-\log (5) \log \left (3+e^4+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^4 + x - Log[5])/(3 + E^4 + x),x]

[Out]

x - Log[5]*Log[3 + E^4 + x]

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fricas [A]  time = 1.01, size = 12, normalized size = 0.50 \begin {gather*} -\log \relax (5) \log \left (x + e^{4} + 3\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(5)+exp(4)+3+x)/(exp(4)+3+x),x, algorithm="fricas")

[Out]

-log(5)*log(x + e^4 + 3) + x

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giac [A]  time = 0.14, size = 13, normalized size = 0.54 \begin {gather*} -\log \relax (5) \log \left ({\left | x + e^{4} + 3 \right |}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(5)+exp(4)+3+x)/(exp(4)+3+x),x, algorithm="giac")

[Out]

-log(5)*log(abs(x + e^4 + 3)) + x

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maple [A]  time = 0.50, size = 13, normalized size = 0.54

method	result	size
default	\(x -\ln \relax (5) \ln \left ({\mathrm e}^{4}+3+x \right )\)	\(13\)
norman	\(x -\ln \relax (5) \ln \left ({\mathrm e}^{4}+3+x \right )\)	\(13\)
risch	\(x -\ln \relax (5) \ln \left ({\mathrm e}^{4}+3+x \right )\)	\(13\)
meijerg	\(-\ln \relax (5) \ln \left (1+\frac {x}{{\mathrm e}^{4}+3}\right )+{\mathrm e}^{4} \ln \left (1+\frac {x}{{\mathrm e}^{4}+3}\right )+3 \ln \left (1+\frac {x}{{\mathrm e}^{4}+3}\right )+\left ({\mathrm e}^{4}+3\right ) \left (\frac {x}{{\mathrm e}^{4}+3}-\ln \left (1+\frac {x}{{\mathrm e}^{4}+3}\right )\right )\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(5)+exp(4)+3+x)/(exp(4)+3+x),x,method=_RETURNVERBOSE)

[Out]

x-ln(5)*ln(exp(4)+3+x)

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maxima [A]  time = 0.36, size = 12, normalized size = 0.50 \begin {gather*} -\log \relax (5) \log \left (x + e^{4} + 3\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(5)+exp(4)+3+x)/(exp(4)+3+x),x, algorithm="maxima")

[Out]

-log(5)*log(x + e^4 + 3) + x

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mupad [B]  time = 8.13, size = 12, normalized size = 0.50 \begin {gather*} x-\ln \relax (5)\,\ln \left (x+{\mathrm {e}}^4+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(4) - log(5) + 3)/(x + exp(4) + 3),x)

[Out]

x - log(5)*log(x + exp(4) + 3)

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sympy [A]  time = 0.24, size = 12, normalized size = 0.50 \begin {gather*} x - \log {\relax (5 )} \log {\left (x + 3 + e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(5)+exp(4)+3+x)/(exp(4)+3+x),x)

[Out]

x - log(5)*log(x + 3 + exp(4))

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