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

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

[Out]

arctan(d*x+c)/d

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {\tan ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (c + d*x)^2)^(-1),x]

[Out]

ArcTan[c + d*x]/d

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fricas [A]  time = 0.85, size = 10, normalized size = 1.00 \[ \frac {\arctan \left (d x + c\right )}{d} \]

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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giac [A]  time = 0.32, size = 10, normalized size = 1.00 \[ \frac {\arctan \left (d x + c\right )}{d} \]

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

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maple [A]  time = 0.00, size = 11, normalized size = 1.10 \[ \frac {\arctan \left (d x +c \right )}{d} \]

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 = 1.57, size = 18, normalized size = 1.80 \[ \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d} \]

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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mupad [B]  time = 0.04, size = 10, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c + d*x)^2 + 1),x)

[Out]

atan(c + d*x)/d

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sympy [C]  time = 0.17, size = 24, normalized size = 2.40 \[ \frac {- \frac {i \log {\left (x + \frac {c - i}{d} \right )}}{2} + \frac {i \log {\left (x + \frac {c + i}{d} \right )}}{2}}{d} \]

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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