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

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 = 0.0237709, 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 \[ \frac{c+d x}{2 d \left ((c+d x)^2+1\right )}+\frac{\tan ^{-1}(c+d x)}{2 d} \]

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)

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Rubi in Sympy [A]  time = 2.46138, size = 26, normalized size = 0.7 \[ \frac{c + d x}{2 d \left (\left (c + d x\right )^{2} + 1\right )} + \frac{\operatorname{atan}{\left (c + d x \right )}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+(d*x+c)**2)**2,x)

[Out]

(c + d*x)/(2*d*((c + d*x)**2 + 1)) + atan(c + d*x)/(2*d)

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Mathematica [A]  time = 0.0194258, 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 = 0.008, size = 59, normalized size = 1.6 \[{\frac{2\,{d}^{2}x+2\,cd}{4\,{d}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}+1 \right ) }}+{\frac{1}{2\,d}\arctan \left ({\frac{2\,{d}^{2}x+2\,cd}{2\,d}} \right ) } \]

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 = 0.907047, size = 69, normalized size = 1.86 \[ \frac{d x + c}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} + 1\right )} d\right )}} + \frac{\arctan \left (\frac{d^{2} x + c d}{d}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 + 1)^(-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 = 0.254473, size = 74, normalized size = 2. \[ \frac{d x +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \arctan \left (d x + c\right ) + c}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} + 1\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 + 1)^(-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 [A]  time = 2.31239, size = 56, normalized size = 1.51 \[ \frac{c + d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} + 2 d} + \frac{- \frac{i \log{\left (x + \frac{c - i}{d} \right )}}{4} + \frac{i \log{\left (x + \frac{c + i}{d} \right )}}{4}}{d} \]

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/XCAS [A]  time = 0.263369, size = 55, normalized size = 1.49 \[ \frac{\arctan \left (d x + c\right )}{2 \, d} + \frac{d x + c}{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 + 1)^(-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)