Integrand size = 21, antiderivative size = 31 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {a \coth ^2(c+d x)}{2 d}+\frac {(a+b) \log (\sinh (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3710, 12, 3556} \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {(a+b) \log (\sinh (c+d x))}{d}-\frac {a \coth ^2(c+d x)}{2 d} \]
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Rule 12
Rule 3556
Rule 3710
Rubi steps \begin{align*} \text {integral}& = -\frac {a \coth ^2(c+d x)}{2 d}+\int (a+b) \coth (c+d x) \, dx \\ & = -\frac {a \coth ^2(c+d x)}{2 d}+(a+b) \int \coth (c+d x) \, dx \\ & = -\frac {a \coth ^2(c+d x)}{2 d}+\frac {(a+b) \log (\sinh (c+d x))}{d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {-a \coth ^2(c+d x)+2 (a+b) (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{2 d} \]
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )+b \ln \left (\sinh \left (d x +c \right )\right )}{d}\) | \(35\) |
default | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )+b \ln \left (\sinh \left (d x +c \right )\right )}{d}\) | \(35\) |
parallelrisch | \(\frac {\left (-2 a -2 b \right ) \ln \left (1-\tanh \left (d x +c \right )\right )+\left (2 a +2 b \right ) \ln \left (\tanh \left (d x +c \right )\right )-\coth \left (d x +c \right )^{2} a -2 \left (a +b \right ) x d}{2 d}\) | \(59\) |
risch | \(-a x -b x -\frac {2 a c}{d}-\frac {2 b c}{d}-\frac {2 a \,{\mathrm e}^{2 d x +2 c}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}\) | \(86\) |
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Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 407, normalized size of antiderivative = 13.13 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {{\left (a + b\right )} d x \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} d x \sinh \left (d x + c\right )^{4} + {\left (a + b\right )} d x - 2 \, {\left ({\left (a + b\right )} d x - a\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} - {\left (a + b\right )} d x + a\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left ({\left (a + b\right )} d x \cosh \left (d x + c\right )^{3} - {\left ({\left (a + b\right )} d x - a\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \]
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\[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \coth ^{3}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (29) = 58\).
Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.42 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac {b \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \]
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Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {{\left (d x + c\right )} {\left (a + b\right )} - {\left (a + b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {2 \, a e^{\left (2 \, d x + 2 \, c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{d} \]
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Time = 1.82 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a+b\right )}{d}-\frac {2\,a}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-x\,\left (a+b\right ) \]
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