3.481 \(\int \frac {1+x^2}{1+x} \, dx\)

Optimal. Leaf size=17 \[ \frac {x^2}{2}-x+2 \log (x+1) \]

[Out]

-x+1/2*x^2+2*ln(1+x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {697} \[ \frac {x^2}{2}-x+2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + x),x]

[Out]

-x + x^2/2 + 2*Log[1 + x]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {1+x^2}{1+x} \, dx &=\int \left (-1+x+\frac {2}{1+x}\right ) \, dx\\ &=-x+\frac {x^2}{2}+2 \log (1+x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 18, normalized size = 1.06 \[ \frac {1}{2} \left (x^2-2 x+4 \log (x+1)-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(1 + x),x]

[Out]

(-3 - 2*x + x^2 + 4*Log[1 + x])/2

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 15, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - x + 2 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(1+x),x, algorithm="fricas")

[Out]

1/2*x^2 - x + 2*log(x + 1)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 16, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} - x + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(1+x),x, algorithm="giac")

[Out]

1/2*x^2 - x + 2*log(abs(x + 1))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {x^{2}}{2}-x +2 \ln \left (x +1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x+1),x)

[Out]

-x+1/2*x^2+2*ln(x+1)

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 15, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - x + 2 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(1+x),x, algorithm="maxima")

[Out]

1/2*x^2 - x + 2*log(x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 15, normalized size = 0.88 \[ 2\,\ln \left (x+1\right )-x+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/(x + 1),x)

[Out]

2*log(x + 1) - x + x^2/2

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 12, normalized size = 0.71 \[ \frac {x^{2}}{2} - x + 2 \log {\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(1+x),x)

[Out]

x**2/2 - x + 2*log(x + 1)

________________________________________________________________________________________