3.81 \(\int \frac{1}{c+(a+b x)^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b \sqrt{c}} \]

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

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Rubi [A]  time = 0.020165, antiderivative size = 21, 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 \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[(c + (a + b*x)^2)^(-1),x]

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

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Rubi in Sympy [A]  time = 2.47644, size = 17, normalized size = 0.81 \[ \frac{\operatorname{atan}{\left (\frac{a + b x}{\sqrt{c}} \right )}}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan((a + b*x)/sqrt(c))/(b*sqrt(c))

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Mathematica [A]  time = 0.00574561, size = 21, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + (a + b*x)^2)^(-1),x]

[Out]

ArcTan[(a + b*x)/Sqrt[c]]/(b*Sqrt[c])

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Maple [A]  time = 0.002, size = 28, normalized size = 1.3 \[{\frac{1}{b}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+(b*x+a)^2),x)

[Out]

1/b/c^(1/2)*arctan(1/2*(2*b^2*x+2*a*b)/b/c^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273865, size = 1, normalized size = 0.05 \[ \left [\frac{\log \left (\frac{2 \, b c x + 2 \, a c +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c\right )} \sqrt{-c}}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b \sqrt{-c}}, \frac{\arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log((2*b*c*x + 2*a*c + (b^2*x^2 + 2*a*b*x + a^2 - c)*sqrt(-c))/(b^2*x^2 + 2
*a*b*x + a^2 + c))/(b*sqrt(-c)), arctan((b*x + a)/sqrt(c))/(b*sqrt(c))]

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Sympy [A]  time = 0.443774, size = 54, normalized size = 2.57 \[ \frac{- \frac{\sqrt{- \frac{1}{c}} \log{\left (x + \frac{a - c \sqrt{- \frac{1}{c}}}{b} \right )}}{2} + \frac{\sqrt{- \frac{1}{c}} \log{\left (x + \frac{a + c \sqrt{- \frac{1}{c}}}{b} \right )}}{2}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+(b*x+a)**2),x)

[Out]

(-sqrt(-1/c)*log(x + (a - c*sqrt(-1/c))/b)/2 + sqrt(-1/c)*log(x + (a + c*sqrt(-1
/c))/b)/2)/b

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GIAC/XCAS [A]  time = 0.26049, size = 23, normalized size = 1.1 \[ \frac{\arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan((b*x + a)/sqrt(c))/(b*sqrt(c))