Optimal. Leaf size=40 \[ a x+\frac{b \log \left (1-(c+d x)^2\right )}{2 d}+\frac{b (c+d x) \tanh ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.023844, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6103, 5910, 260} \[ a x+\frac{b \log \left (1-(c+d x)^2\right )}{2 d}+\frac{b (c+d x) \tanh ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 6103
Rule 5910
Rule 260
Rubi steps
\begin{align*} \int \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=a x+b \int \tanh ^{-1}(c+d x) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac{b (c+d x) \tanh ^{-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) \tanh ^{-1}(c+d x)}{d}+\frac{b \log \left (1-(c+d x)^2\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0156735, size = 48, normalized size = 1.2 \[ a x+\frac{b ((c+1) \log (c+d x+1)-(c-1) \log (-c-d x+1))}{2 d}+b x \tanh ^{-1}(c+d x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 44, normalized size = 1.1 \begin{align*} ax+b{\it Artanh} \left ( dx+c \right ) x+{\frac{b{\it Artanh} \left ( dx+c \right ) c}{d}}+{\frac{b\ln \left ( 1- \left ( dx+c \right ) ^{2} \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944952, size = 49, normalized size = 1.22 \begin{align*} a x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94325, size = 158, normalized size = 3.95 \begin{align*} \frac{b d x \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 2 \, a d x +{\left (b c + b\right )} \log \left (d x + c + 1\right ) -{\left (b c - b\right )} \log \left (d x + c - 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.775077, size = 46, normalized size = 1.15 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{atanh}{\left (c + d x \right )}}{d} + x \operatorname{atanh}{\left (c + d x \right )} + \frac{\log{\left (c + d x + 1 \right )}}{d} - \frac{\operatorname{atanh}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \operatorname{atanh}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17558, size = 82, normalized size = 2.05 \begin{align*} \frac{1}{2} \,{\left (d{\left (\frac{{\left (c + 1\right )} \log \left ({\left | d x + c + 1 \right |}\right )}{d^{2}} - \frac{{\left (c - 1\right )} \log \left ({\left | d x + c - 1 \right |}\right )}{d^{2}}\right )} + x \log \left (-\frac{d x + c + 1}{d x + c - 1}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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