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

Optimal. Leaf size=37 \[ \frac{c+d x}{2 d \left ((c+d x)^2+1\right )}+\frac{\tan ^{-1}(c+d x)}{2 d} \]

[Out]

(c + d*x)/(2*d*(1 + (c + d*x)^2)) + ArcTan[c + d*x]/(2*d)

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Rubi [A]  time = 0.0095384, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 199, 203} \[ \frac{c+d x}{2 d \left ((c+d x)^2+1\right )}+\frac{\tan ^{-1}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + (c + d*x)^2)^(-2),x]

[Out]

(c + d*x)/(2*d*(1 + (c + d*x)^2)) + ArcTan[c + d*x]/(2*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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.011791, size = 31, normalized size = 0.84 \[ \frac{\frac{c+d x}{(c+d x)^2+1}+\tan ^{-1}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (c + d*x)^2)^(-2),x]

[Out]

((c + d*x)/(1 + (c + d*x)^2) + ArcTan[c + d*x])/(2*d)

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Maple [A]  time = 0.004, size = 59, normalized size = 1.6 \begin{align*}{\frac{2\,{d}^{2}x+2\,cd}{4\,{d}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}+1 \right ) }}+{\frac{1}{2\,d}\arctan \left ({\frac{2\,{d}^{2}x+2\,cd}{2\,d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*d^2*x+2*c*d)/d^2/(d^2*x^2+2*c*d*x+c^2+1)+1/2/d*arctan(1/2*(2*d^2*x+2*c*d)/d)

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Maxima [A]  time = 1.45754, size = 69, normalized size = 1.86 \begin{align*} \frac{d x + c}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} + 1\right )} d\right )}} + \frac{\arctan \left (\frac{d^{2} x + c d}{d}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(d*x + c)/(d^3*x^2 + 2*c*d^2*x + (c^2 + 1)*d) + 1/2*arctan((d^2*x + c*d)/d)/d

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Fricas [A]  time = 1.80018, size = 134, normalized size = 3.62 \begin{align*} \frac{d x +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \arctan \left (d x + c\right ) + c}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} + 1\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(d*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)*arctan(d*x + c) + c)/(d^3*x^2 + 2*c*d^2*x + (c^2 + 1)*d)

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Sympy [C]  time = 0.58651, size = 56, normalized size = 1.51 \begin{align*} \frac{c + d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} + 2 d} + \frac{- \frac{i \log{\left (x + \frac{c - i}{d} \right )}}{4} + \frac{i \log{\left (x + \frac{c + i}{d} \right )}}{4}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c + d*x)/(2*c**2*d + 4*c*d**2*x + 2*d**3*x**2 + 2*d) + (-I*log(x + (c - I)/d)/4 + I*log(x + (c + I)/d)/4)/d

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Giac [A]  time = 1.1549, size = 55, normalized size = 1.49 \begin{align*} \frac{\arctan \left (d x + c\right )}{2 \, d} + \frac{d x + c}{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(d*x + c)/d + 1/2*(d*x + c)/((d^2*x^2 + 2*c*d*x + c^2 + 1)*d)