Optimal. Leaf size=10 \[ \frac{\tan ^{-1}(c+d x)}{d} \]
[Out]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]