Integrand size = 21, antiderivative size = 49 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+\frac {(a+b) \log (\sinh (c+d x))}{d} \]
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Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3710, 12, 3554, 3556} \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {(a+b) \coth ^2(c+d x)}{2 d}+\frac {(a+b) \log (\sinh (c+d x))}{d}-\frac {a \coth ^4(c+d x)}{4 d} \]
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Rule 12
Rule 3554
Rule 3556
Rule 3710
Rubi steps \begin{align*} \text {integral}& = -\frac {a \coth ^4(c+d x)}{4 d}+\int (a+b) \coth ^3(c+d x) \, dx \\ & = -\frac {a \coth ^4(c+d x)}{4 d}+(a+b) \int \coth ^3(c+d x) \, dx \\ & = -\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+(a+b) \int \coth (c+d x) \, dx \\ & = -\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+\frac {(a+b) \log (\sinh (c+d x))}{d} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {2 (a+b) \coth ^2(c+d x)+a \coth ^4(c+d x)-4 (a+b) (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{4 d} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(56\) |
default | \(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(56\) |
parallelrisch | \(\frac {\left (-4 a -4 b \right ) \ln \left (1-\tanh \left (d x +c \right )\right )+\left (4 a +4 b \right ) \ln \left (\tanh \left (d x +c \right )\right )-\coth \left (d x +c \right )^{4} a +\left (-2 a -2 b \right ) \coth \left (d x +c \right )^{2}-4 \left (a +b \right ) x d}{4 d}\) | \(75\) |
risch | \(-a x -b x -\frac {2 a c}{d}-\frac {2 b c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (2 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}-2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+2 a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}\) | \(137\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (45) = 90\).
Time = 0.27 (sec) , antiderivative size = 1216, normalized size of antiderivative = 24.82 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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\[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \coth ^{5}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (45) = 90\).
Time = 0.19 (sec) , antiderivative size = 206, normalized size of antiderivative = 4.20 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (45) = 90\).
Time = 0.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.90 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {{\left (d x + c\right )} {\left (a + b\right )} - {\left (a + b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {2 \, {\left ({\left (2 \, a + b\right )} e^{\left (6 \, d x + 6 \, c\right )} - 2 \, {\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (2 \, a + b\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{d} \]
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Time = 1.83 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.61 \[ \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a+b\right )}{d}-\frac {2\,\left (2\,a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-x\,\left (a+b\right )-\frac {2\,\left (4\,a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,a}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \]
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