3.1.89 \(\int \frac {1}{1+(c+d x)^2} \, dx\) [89]

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 = {253, 209} \begin {gather*} \frac {\text {ArcTan}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (c + d*x)^2)^(-1),x]

[Out]

ArcTan[c + d*x]/d

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 253

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 {\text {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 = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}(c+d x)}{d} \end {gather*}

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 = 0.28, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {\arctan \left (d x +c \right )}{d}\)	\(11\)
risch	\(\frac {\arctan \left (d x +c \right )}{d}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(d*x+c)/d

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Maxima [A]
time = 0.47, size = 18, normalized size = 1.80 \begin {gather*} \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d} \end {gather*}

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 = 0.38, size = 10, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (d x + c\right )}{d} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.06, size = 24, normalized size = 2.40 \begin {gather*} \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 {gather*}

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 = 5.84, size = 10, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (d x + c\right )}{d} \end {gather*}

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

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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