Integrand size = 19, antiderivative size = 25 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {b \log (\cosh (c+d x))}{d}+\frac {a \log (\sinh (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3706, 3556} \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {a \log (\sinh (c+d x))}{d}+\frac {b \log (\cosh (c+d x))}{d} \]
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Rule 3556
Rule 3706
Rubi steps \begin{align*} \text {integral}& = a \int \coth (c+d x) \, dx+b \int \tanh (c+d x) \, dx \\ & = \frac {b \log (\cosh (c+d x))}{d}+\frac {a \log (\sinh (c+d x))}{d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {b \log (\cosh (c+d x))}{d}+\frac {a (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{d} \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a \ln \left (\sinh \left (d x +c \right )\right )+b \ln \left (\cosh \left (d x +c \right )\right )}{d}\) | \(24\) |
default | \(\frac {a \ln \left (\sinh \left (d x +c \right )\right )+b \ln \left (\cosh \left (d x +c \right )\right )}{d}\) | \(24\) |
parallelrisch | \(\frac {\left (-a -b \right ) \ln \left (1-\tanh \left (d x +c \right )\right )+a \ln \left (\tanh \left (d x +c \right )\right )-\left (a +b \right ) x d}{d}\) | \(41\) |
risch | \(-a x -b x -\frac {2 b c}{d}-\frac {2 a c}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) b}{d}+\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {{\left (a + b\right )} d x - b \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - a \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{d} \]
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\[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \coth {\left (c + d x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {b \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac {a \log \left (\sinh \left (d x + c\right )\right )}{d} \]
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none
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {{\left (d x + c\right )} {\left (a + b\right )} - b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - a \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{d} \]
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Time = 1.85 (sec) , antiderivative size = 228, normalized size of antiderivative = 9.12 \[ \int \coth (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {a\,\ln \left (8\,a\,b-4\,a^2-4\,b^2+4\,a^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-8\,a\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d}-b\,x-\frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^2-2\,a\,b+b^2}}-\frac {b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^2-2\,a\,b+b^2}}\right )\,\sqrt {a^2-2\,a\,b+b^2}}{\sqrt {-d^2}}-a\,x+\frac {b\,\ln \left (8\,a\,b-4\,a^2-4\,b^2+4\,a^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-8\,a\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d} \]
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