Optimal. Leaf size=38 \[ a x+\frac {b (c+d x) \text {ArcTan}(c+d x)}{d}-\frac {b \log \left (1+(c+d x)^2\right )}{2 d} \]
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Rubi [A]
time = 0.01, 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 = {5147, 4930,
266} \begin {gather*} a x+\frac {b (c+d x) \text {ArcTan}(c+d x)}{d}-\frac {b \log \left ((c+d x)^2+1\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4930
Rule 5147
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}(c+d x)\right ) \, dx &=a x+b \int \tan ^{-1}(c+d x) \, dx\\ &=a x+\frac {b \text {Subst}\left (\int \tan ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \tan ^{-1}(c+d x)}{d}-\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \tan ^{-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 \begin {gather*} a x+b x \text {ArcTan}(c+d x)-\frac {b \left (-2 c \text {ArcTan}(c+d x)+\log \left (1+c^2+2 c d x+d^2 x^2\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 42, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \left (d x +c \right ) \arctan \left (d x +c \right )-\frac {b \ln \left (1+\left (d x +c \right )^{2}\right )}{2}}{d}\) | \(39\) |
default | \(a x +b \arctan \left (d x +c \right ) x +\frac {b \arctan \left (d x +c \right ) c}{d}-\frac {b \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d}\) | \(42\) |
risch | \(a x -\frac {i b x \ln \left (1+i \left (d x +c \right )\right )}{2}+\frac {i b x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {b \arctan \left (d x +c \right ) c}{d}-\frac {b \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.95 \begin {gather*} a x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 48, normalized size = 1.26 \begin {gather*} \frac {2 \, a d x + 2 \, {\left (b d x + b c\right )} \arctan \left (d x + c\right ) - b \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 51, normalized size = 1.34 \begin {gather*} a x + b \left (\begin {cases} \frac {c \operatorname {atan}{\left (c + d x \right )}}{d} + x \operatorname {atan}{\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 {atan}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 36, normalized size = 0.95 \begin {gather*} a x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 49, normalized size = 1.29 \begin {gather*} a\,x+b\,x\,\mathrm {atan}\left (c+d\,x\right )-\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{2\,d}+\frac {b\,c\,\mathrm {atan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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