3.89 \(\int \frac{1}{1+(c+d x)^2} \, dx\)

Optimal. Leaf size=10 \[ \frac{\tan ^{-1}(c+d x)}{d} \]

[Out]

ArcTan[c + d*x]/d

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Rubi [A]  time = 0.002813, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {247, 203} \[ \frac{\tan ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + (c + d*x)^2)^(-1),x]

[Out]

ArcTan[c + d*x]/d

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{1+(c+d x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\tan ^{-1}(c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0044184, size = 10, normalized size = 1. \[ \frac{\tan ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (c + d*x)^2)^(-1),x]

[Out]

ArcTan[c + d*x]/d

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Maple [A]  time = 0.003, size = 11, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+(d*x+c)^2),x)

[Out]

arctan(d*x+c)/d

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Maxima [A]  time = 1.5606, size = 24, normalized size = 2.4 \begin{align*} \frac{\arctan \left (\frac{d^{2} x + c d}{d}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan((d^2*x + c*d)/d)/d

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Fricas [A]  time = 1.68576, size = 26, normalized size = 2.6 \begin{align*} \frac{\arctan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(d*x + c)/d

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Sympy [C]  time = 0.158548, size = 24, normalized size = 2.4 \begin{align*} \frac{- \frac{i \log{\left (x + \frac{c - i}{d} \right )}}{2} + \frac{i \log{\left (x + \frac{c + i}{d} \right )}}{2}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*log(x + (c - I)/d)/2 + I*log(x + (c + I)/d)/2)/d

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Giac [A]  time = 1.12743, size = 14, normalized size = 1.4 \begin{align*} \frac{\arctan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(d*x + c)/d