Integrand size = 14, antiderivative size = 31 \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt [3]{b \coth ^3(c+d x)}} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3739, 3556} \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt [3]{b \coth ^3(c+d x)}} \]
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Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (c+d x) \int \tanh (c+d x) \, dx}{\sqrt [3]{b \coth ^3(c+d x)}} \\ & = \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt [3]{b \coth ^3(c+d x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt [3]{b \coth ^3(c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(191\) vs. \(2(29)=58\).
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 6.19
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right ) x}{{\left (\frac {b \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )}^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 \left ({\mathrm e}^{2 d x +2 c}+1\right ) \left (d x +c \right )}{{\left (\frac {b \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )}^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right ) d}+\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{{\left (\frac {b \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )}^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right ) d}\) | \(192\) |
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=-\frac {{\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x e^{\left (2 \, d x + 2 \, c\right )} + d x - {\left (e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}}}{b d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b d e^{\left (2 \, d x + 2 \, c\right )} + b d} \]
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\[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{b \coth ^{3}{\left (c + d x \right )}}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\frac {d x + c}{b^{\frac {1}{3}} d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{\frac {1}{3}} d} \]
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\[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{3}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{b \coth ^3(c+d x)}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{1/3}} \,d x \]
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