Optimal. Leaf size=33 \[ \frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5039, 4846, 260} \[ \frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 5039
Rubi steps
\begin {align*} \int \tan ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \tan ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \tan ^{-1}(a+b x)}{b}-\frac {\log \left (1+(a+b x)^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.18 \[ -\frac {\log \left (a^2+2 a b x+b^2 x^2+1\right )-2 (a+b x) \tan ^{-1}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 39, normalized size = 1.18 \[ \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 31, normalized size = 0.94 \[ \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.09 \[ x \arctan \left (b x +a \right )+\frac {\arctan \left (b x +a \right ) a}{b}-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 31, normalized size = 0.94 \[ \frac {2 \, {\left (b x + a\right )} \arctan \left (b x + a\right ) - \log \left ({\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 42, normalized size = 1.27 \[ x\,\mathrm {atan}\left (a+b\,x\right )-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )-2\,a\,\mathrm {atan}\left (a+b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 46, normalized size = 1.39 \[ \begin {cases} \frac {a \operatorname {atan}{\left (a + b x \right )}}{b} + x \operatorname {atan}{\left (a + b x \right )} - \frac {\log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {atan}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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