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 = 0.00874482, 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 \[ \frac{\tan ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (c + d*x)^2)^(-1),x]

[Out]

ArcTan[c + d*x]/d

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Rubi in Sympy [A]  time = 1.62704, size = 7, normalized size = 0.7 \[ \frac{\operatorname{atan}{\left (c + d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(c + d*x)/d

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Mathematica [A]  time = 0.00631198, 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 = 0.005, size = 11, normalized size = 1.1 \[{\frac{\arctan \left ( dx+c \right ) }{d}} \]

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 = 0.902752, size = 24, normalized size = 2.4 \[ \frac{\arctan \left (\frac{d^{2} x + c d}{d}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan((d^2*x + c*d)/d)/d

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Fricas [A]  time = 0.252228, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(d*x + c)/d

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Sympy [A]  time = 0.385454, size = 24, normalized size = 2.4 \[ \frac{- \frac{i \log{\left (x + \frac{c - i}{d} \right )}}{2} + \frac{i \log{\left (x + \frac{c + i}{d} \right )}}{2}}{d} \]

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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GIAC/XCAS [A]  time = 0.259758, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(d*x + c)/d