Integrand size = 12, antiderivative size = 132 \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d} \]
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Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3557, 335, 281, 298, 31, 648, 631, 210, 642} \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{2/3} (b \coth (c+d x))^{2/3}+b^{4/3}+(b \coth (c+d x))^{4/3}\right )}{4 d} \]
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 335
Rule 631
Rule 642
Rule 648
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d} \\ & = -\frac {(3 b) \text {Subst}\left (\int \frac {x^3}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d} \\ & = -\frac {(3 b) \text {Subst}\left (\int \frac {x}{-b^2+x^3} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d} \\ & = -\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-b^{2/3}+x} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {-b^{2/3}+x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{2 d} \\ & = -\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {b^{2/3}+2 x}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 d}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{b^{4/3}+b^{2/3} x+x^2} \, dx,x,(b \coth (c+d x))^{2/3}\right )}{4 d} \\ & = -\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d}+\frac {\left (3 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 (b \coth (c+d x))^{2/3}}{b^{2/3}}\right )}{2 d} \\ & = -\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1+\frac {2 (b \coth (c+d x))^{2/3}}{b^{2/3}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=-\frac {(b \coth (c+d x))^{4/3} \left (\log \left (1-\sqrt [3]{\coth ^2(c+d x)}\right )-\sqrt [3]{-1} \log \left (1+\sqrt [3]{-1} \sqrt [3]{\coth ^2(c+d x)}\right )+(-1)^{2/3} \log \left (1-(-1)^{2/3} \sqrt [3]{\coth ^2(c+d x)}\right )\right )}{2 b d \coth ^2(c+d x)^{2/3}} \]
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Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\) | \(109\) |
default | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\) | \(109\) |
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (99) = 198\).
Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.20 \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=-\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}}{3 \, b}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (-\left (-b\right )^{\frac {2}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + \left (-b\right )^{\frac {1}{3}} \log \left (\frac {{\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \left (-b\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \left (-b\right )^{\frac {1}{3}} + {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b\right )} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}\right )}{4 \, d} \]
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\[ \int \sqrt [3]{b \coth (c+d x)} \, dx=\int \sqrt [3]{b \coth {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt [3]{b \coth (c+d x)} \, dx=\int { \left (b \coth \left (d x + c\right )\right )^{\frac {1}{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (99) = 198\).
Time = 0.43 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.64 \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=-\frac {b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2}} - \frac {{\left | b \right |}^{\frac {4}{3}} \log \left (\left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}} + \frac {{\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )}{b^{2}} + \frac {2 \, {\left | b \right |}^{\frac {4}{3}} \log \left ({\left | \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}} \right |}\right )}{b^{2}}\right )}}{4 \, d} \]
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Time = 2.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11 \[ \int \sqrt [3]{b \coth (c+d x)} \, dx=\frac {{\left (-b\right )}^{1/3}\,\ln \left (81\,{\left (-b\right )}^{16/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}-81\,b^6\right )}{2\,d}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (-\frac {81\,b^6}{d^4}-\frac {81\,{\left (-b\right )}^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (-\frac {81\,b^6}{d^4}+\frac {162\,{\left (-b\right )}^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]
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