\(\int \frac {b+a x}{1+x^2} \, dx\) [9]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {b+a x}{1+x^2} \, dx=b \arctan (x)+\frac {1}{2} a \log \left (1+x^2\right ) \]

[Out]

b*arctan(x)+1/2*a*ln(x^2+1)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {649, 209, 266} \[ \int \frac {b+a x}{1+x^2} \, dx=\frac {1}{2} a \log \left (x^2+1\right )+b \arctan (x) \]

[In]

Int[(b + a*x)/(1 + x^2),x]

[Out]

b*ArcTan[x] + (a*Log[1 + x^2])/2

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rubi steps \begin{align*} \text {integral}& = a \int \frac {x}{1+x^2} \, dx+b \int \frac {1}{1+x^2} \, dx \\ & = b \arctan (x)+\frac {1}{2} a \log \left (1+x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {b+a x}{1+x^2} \, dx=b \arctan (x)+\frac {1}{2} a \log \left (1+x^2\right ) \]

[In]

Integrate[(b + a*x)/(1 + x^2),x]

[Out]

b*ArcTan[x] + (a*Log[1 + x^2])/2

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(b \arctan \left (x \right )+\frac {a \ln \left (x^{2}+1\right )}{2}\)	\(15\)
meijerg	\(b \arctan \left (x \right )+\frac {a \ln \left (x^{2}+1\right )}{2}\)	\(15\)
risch	\(b \arctan \left (x \right )+\frac {a \ln \left (x^{2}+1\right )}{2}\)	\(15\)
parallelrisch	\(\frac {\ln \left (x -i\right ) a}{2}-\frac {i \ln \left (x -i\right ) b}{2}+\frac {\ln \left (x +i\right ) a}{2}+\frac {i \ln \left (x +i\right ) b}{2}\)	\(36\)

[In]

int((a*x+b)/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

b*arctan(x)+1/2*a*ln(x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x}{1+x^2} \, dx=b \arctan \left (x\right ) + \frac {1}{2} \, a \log \left (x^{2} + 1\right ) \]

[In]

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

[Out]

b*arctan(x) + 1/2*a*log(x^2 + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {b+a x}{1+x^2} \, dx=\left (\frac {a}{2} - \frac {i b}{2}\right ) \log {\left (x - i \right )} + \left (\frac {a}{2} + \frac {i b}{2}\right ) \log {\left (x + i \right )} \]

[In]

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

[Out]

(a/2 - I*b/2)*log(x - I) + (a/2 + I*b/2)*log(x + I)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x}{1+x^2} \, dx=b \arctan \left (x\right ) + \frac {1}{2} \, a \log \left (x^{2} + 1\right ) \]

[In]

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

[Out]

b*arctan(x) + 1/2*a*log(x^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x}{1+x^2} \, dx=b \arctan \left (x\right ) + \frac {1}{2} \, a \log \left (x^{2} + 1\right ) \]

[In]

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

[Out]

b*arctan(x) + 1/2*a*log(x^2 + 1)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {b+a x}{1+x^2} \, dx=\frac {a\,\ln \left (x^2+1\right )}{2}+b\,\mathrm {atan}\left (x\right ) \]

[In]

int((b + a*x)/(x^2 + 1),x)

[Out]

(a*log(x^2 + 1))/2 + b*atan(x)