Integrand size = 10, antiderivative size = 40 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a x+\frac {b (c+d x) \coth ^{-1}(c+d x)}{d}+\frac {b \log \left (1-(c+d x)^2\right )}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6239, 6022, 266} \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a x+\frac {b \log \left (1-(c+d x)^2\right )}{2 d}+\frac {b (c+d x) \coth ^{-1}(c+d x)}{d} \]
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Rule 266
Rule 6022
Rule 6239
Rubi steps \begin{align*} \text {integral}& = a x+b \int \coth ^{-1}(c+d x) \, dx \\ & = a x+\frac {b \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d} \\ & = a x+\frac {b (c+d x) \coth ^{-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) \coth ^{-1}(c+d x)}{d}+\frac {b \log \left (1-(c+d x)^2\right )}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a x+b x \coth ^{-1}(c+d x)+\frac {b (-((-1+c) \log (1-c-d x))+(1+c) \log (1+c+d x))}{2 d} \]
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Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88
method | result | size |
default | \(a x +\frac {b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(35\) |
parts | \(a x +\frac {b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(35\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(40\) |
parallelrisch | \(-\frac {b \left (-x \,\operatorname {arccoth}\left (d x +c \right ) d^{2}-\operatorname {arccoth}\left (d x +c \right ) c d -\ln \left (d x +c -1\right ) d -\operatorname {arccoth}\left (d x +c \right ) d \right )}{d^{2}}+a x\) | \(53\) |
risch | \(a x +\frac {b x \ln \left (d x +c +1\right )}{2}-\frac {b x \ln \left (d x +c -1\right )}{2}-\frac {b \ln \left (d x +c -1\right ) c}{2 d}+\frac {b \ln \left (-d x -c -1\right ) c}{2 d}+\frac {b \ln \left (d x +c -1\right )}{2 d}+\frac {b \ln \left (-d x -c -1\right )}{2 d}\) | \(87\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.50 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {b d x \log \left (\frac {d x + c + 1}{d x + c - 1}\right ) + 2 \, a d x + {\left (b c + b\right )} \log \left (d x + c + 1\right ) - {\left (b c - b\right )} \log \left (d x + c - 1\right )}{2 \, d} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a x + b \left (\begin {cases} \frac {c \operatorname {acoth}{\left (c + d x \right )}}{d} + x \operatorname {acoth}{\left (c + d x \right )} + \frac {\log {\left (c + d x + 1 \right )}}{d} - \frac {\operatorname {acoth}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \operatorname {acoth}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (38) = 76\).
Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 5.05 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} b {\left (\frac {\log \left (\frac {{\left | d x + c + 1 \right |}}{{\left | d x + c - 1 \right |}}\right )}{d^{2}} - \frac {\log \left ({\left | \frac {d x + c + 1}{d x + c - 1} - 1 \right |}\right )}{d^{2}} + \frac {\log \left (-\frac {\frac {1}{c - \frac {{\left (\frac {{\left (d x + c + 1\right )} {\left (c - 1\right )}}{d x + c - 1} - c - 1\right )} d}{\frac {{\left (d x + c + 1\right )} d}{d x + c - 1} - d}} + 1}{\frac {1}{c - \frac {{\left (\frac {{\left (d x + c + 1\right )} {\left (c - 1\right )}}{d x + c - 1} - c - 1\right )} d}{\frac {{\left (d x + c + 1\right )} d}{d x + c - 1} - d}} - 1}\right )}{d^{2} {\left (\frac {d x + c + 1}{d x + c - 1} - 1\right )}}\right )} + a x \]
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Time = 4.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a\,x+\frac {\frac {b\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2}+b\,c\,\mathrm {acoth}\left (c+d\,x\right )}{d}+b\,x\,\mathrm {acoth}\left (c+d\,x\right ) \]
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