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 = 0.02, antiderivative size = 40, 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 = {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 260
Rule 5910
Rule 6103
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*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.20 \[ a x+\frac {b ((c+1) \log (c+d x+1)-(c-1) \log (-c-d x+1))}{2 d}+b x \tanh ^{-1}(c+d x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 61, normalized size = 1.52 \[ \frac {b d x \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 2 \, a d x + {\left (b c + b\right )} \log \left (d x + c + 1\right ) - {\left (b c - b\right )} \log \left (d x + c - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 200, normalized size = 5.00 \[ \frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} b {\left (\frac {\log \left (\frac {{\left | -d x - c - 1 \right |}}{{\left | d x + c - 1 \right |}}\right )}{d^{2}} - \frac {\log \left ({\left | -\frac {d x + c + 1}{d x + c - 1} + 1 \right |}\right )}{d^{2}} + \frac {\log \left (-\frac {c - \frac {{\left (\frac {{\left (d x + c + 1\right )} {\left (c - 1\right )}}{d x + c - 1} - c - 1\right )} d}{\frac {{\left (d x + c + 1\right )} d}{d x + c - 1} - d} + 1}{c - \frac {{\left (\frac {{\left (d x + c + 1\right )} {\left (c - 1\right )}}{d x + c - 1} - c - 1\right )} d}{\frac {{\left (d x + c + 1\right )} d}{d x + c - 1} - d} - 1}\right )}{d^{2} {\left (\frac {d x + c + 1}{d x + c - 1} - 1\right )}}\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 44, normalized size = 1.10 \[ a x +b \arctanh \left (d x +c \right ) x +\frac {b \arctanh \left (d x +c \right ) c}{d}+\frac {b \ln \left (1-\left (d x +c \right )^{2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 36, normalized size = 0.90 \[ a x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 48, normalized size = 1.20 \[ a\,x+\frac {\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2}+b\,c\,\mathrm {atanh}\left (c+d\,x\right )}{d}+b\,x\,\mathrm {atanh}\left (c+d\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 46, normalized size = 1.15 \[ a x + b \left (\begin {cases} \frac {c \operatorname {atanh}{\left (c + d x \right )}}{d} + x \operatorname {atanh}{\left (c + d x \right )} + \frac {\log {\left (c + d x + 1 \right )}}{d} - \frac {\operatorname {atanh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \operatorname {atanh}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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