Optimal. Leaf size=21 \[ \frac {\tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {253, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{c+(a+b x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 28, normalized size = 1.33
method | result | size |
default | \(\frac {\arctan \left (\frac {2 b^{2} x +2 a b}{2 b \sqrt {c}}\right )}{b \sqrt {c}}\) | \(28\) |
risch | \(-\frac {\ln \left (b x +\sqrt {-c}+a \right )}{2 \sqrt {-c}\, b}+\frac {\ln \left (-b x +\sqrt {-c}-a \right )}{2 \sqrt {-c}\, b}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 24, normalized size = 1.14 \begin {gather*} \frac {\arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 83, normalized size = 3.95 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b c}, \frac {\arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b \sqrt {c}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (17) = 34\).
time = 0.07, size = 54, normalized size = 2.57 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.02, size = 17, normalized size = 0.81 \begin {gather*} \frac {\arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 17, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atan}\left (\frac {a+b\,x}{\sqrt {c}}\right )}{b\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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