3.18.78 \(\int \frac {3+x}{1+x} \, dx\)

Optimal. Leaf size=13 \[ \log \left (-\frac {25}{2} e^x (1+x)^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} x+2 \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + x)/(1 + x),x]

[Out]

x + 2*Log[1 + 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 {2}{1+x}\right ) \, dx\\ &=x+2 \log (1+x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 8, normalized size = 0.62 \begin {gather*} x+2 \log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/(1 + x),x]

[Out]

x + 2*Log[1 + x]

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 8, normalized size = 0.62 \begin {gather*} x + 2 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x+1),x, algorithm="fricas")

[Out]

x + 2*log(x + 1)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 9, normalized size = 0.69 \begin {gather*} x + 2 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x+1),x, algorithm="giac")

[Out]

x + 2*log(abs(x + 1))

________________________________________________________________________________________

maple [A]  time = 0.31, size = 9, normalized size = 0.69

method	result	size
default	\(x +2 \ln \left (x +1\right )\)	\(9\)
norman	\(x +2 \ln \left (x +1\right )\)	\(9\)
meijerg	\(x +2 \ln \left (x +1\right )\)	\(9\)
risch	\(x +2 \ln \left (x +1\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)/(x+1),x,method=_RETURNVERBOSE)

[Out]

x+2*ln(x+1)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 8, normalized size = 0.62 \begin {gather*} x + 2 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x+1),x, algorithm="maxima")

[Out]

x + 2*log(x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 8, normalized size = 0.62 \begin {gather*} x+2\,\ln \left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(x + 1),x)

[Out]

x + 2*log(x + 1)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 7, normalized size = 0.54 \begin {gather*} x + 2 \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x+1),x)

[Out]

x + 2*log(x + 1)

________________________________________________________________________________________