Optimal. Leaf size=38 \[ a x+\frac {b \log \left ((c+d x)^2+1\right )}{2 d}+\frac {b (c+d x) \cot ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5040, 4847, 260} \[ a x+\frac {b \log \left ((c+d x)^2+1\right )}{2 d}+\frac {b (c+d x) \cot ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rule 5040
Rubi steps
\begin {align*} \int \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=a x+b \int \cot ^{-1}(c+d x) \, dx\\ &=a x+\frac {b \operatorname {Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \cot ^{-1}(c+d x)}{d}+\frac {b \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \cot ^{-1}(c+d x)}{d}+\frac {b \log \left (1+(c+d x)^2\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.29 \[ a x+\frac {b \left (\log \left (c^2+2 c d x+d^2 x^2+1\right )-2 c \tan ^{-1}(c+d x)\right )}{2 d}+b x \cot ^{-1}(c+d x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 52, normalized size = 1.37 \[ \frac {2 \, b d x \operatorname {arccot}\left (d x + c\right ) + 2 \, a d x - 2 \, b c \arctan \left (d x + c\right ) + b \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 116, normalized size = 3.05 \[ a x - \frac {{\left (\arctan \left (\frac {1}{d x + c}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right )^{2} + \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right ) - \arctan \left (\frac {1}{d x + c}\right )\right )} b}{2 \, d \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{d x + c}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 1.11 \[ a x +b \,\mathrm {arccot}\left (d x +c \right ) x +\frac {b \,\mathrm {arccot}\left (d x +c \right ) c}{d}+\frac {b \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 0.89 \[ a x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 48, normalized size = 1.26 \[ a\,x+\frac {\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{2}+b\,c\,\mathrm {acot}\left (c+d\,x\right )}{d}+b\,x\,\mathrm {acot}\left (c+d\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 51, normalized size = 1.34 \[ a x + b \left (\begin {cases} \frac {c \operatorname {acot}{\left (c + d x \right )}}{d} + x \operatorname {acot}{\left (c + d x \right )} + \frac {\log {\left (c^{2} + 2 c d x + d^{2} x^{2} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \operatorname {acot}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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