Integrand size = 12, antiderivative size = 129 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=\frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}-\frac {b \left (3 a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^3 d} \]
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Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3564, 3610, 3612, 3611} \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=\frac {2 a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}-\frac {b \left (3 a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^3}+\frac {a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3} \]
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Rule 3564
Rule 3610
Rule 3611
Rule 3612
Rubi steps \begin{align*} \text {integral}& = \frac {b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac {\int \frac {a-b \coth (c+d x)}{(a+b \coth (c+d x))^2} \, dx}{a^2-b^2} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}+\frac {\int \frac {a^2+b^2-2 a b \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^2} \\ & = \frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}-\frac {\left (i b \left (3 a^2+b^2\right )\right ) \int \frac {-i b-i a \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^3} \\ & = \frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}-\frac {b \left (3 a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^3 d} \\ \end{align*}
Time = 3.56 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=-\frac {\frac {\log (1-\tanh (c+d x))}{(a+b)^3}-\frac {\log (1+\tanh (c+d x))}{(a-b)^3}+\frac {b \left (2 \left (3 a^2+b^2\right ) \log (b+a \tanh (c+d x))+\frac {b \left (-a^2+b^2\right ) \left (-5 a^2 b+b^3+\left (-6 a^3+2 a b^2\right ) \tanh (c+d x)\right )}{a^2 (b+a \tanh (c+d x))^2}\right )}{\left (a^2-b^2\right )^3}}{2 d} \]
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Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {\frac {b}{2 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{3}}}{d}\) | \(130\) |
default | \(\frac {\frac {b}{2 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{3}}}{d}\) | \(130\) |
parallelrisch | \(\frac {-3 \left (a^{2}+\frac {b^{2}}{3}\right ) b \,a^{2} \left (b +a \tanh \left (d x +c \right )\right )^{2} \ln \left (b +a \tanh \left (d x +c \right )\right )+3 \left (a^{2}+\frac {b^{2}}{3}\right ) b \,a^{2} \left (b +a \tanh \left (d x +c \right )\right )^{2} \ln \left (1-\tanh \left (d x +c \right )\right )+\left (a^{4} d x \left (a +b \right )^{2} \tanh \left (d x +c \right )^{2}+\left (2 a^{5} b d x +b^{2} \left (4 d x -3\right ) a^{4}+b^{3} \left (2 d x +3\right ) a^{3}+a^{2} b^{4}-a \,b^{5}\right ) \tanh \left (d x +c \right )+b^{2} \left (a^{4} d x +b \left (2 d x -\frac {5}{2}\right ) a^{3}+b^{2} \left (d x +\frac {5}{2}\right ) a^{2}+\frac {a \,b^{3}}{2}-\frac {b^{4}}{2}\right )\right ) \left (a +b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} d \,a^{2} \left (b +a \tanh \left (d x +c \right )\right )^{2}}\) | \(233\) |
risch | \(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}+\frac {6 b \,a^{2} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 b^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {6 b c \,a^{2}}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b^{3} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 b^{2} \left (3 a^{2} {\mathrm e}^{2 d x +2 c}+2 a b \,{\mathrm e}^{2 d x +2 c}-{\mathrm e}^{2 d x +2 c} b^{2}-3 a^{2}+3 a b \right )}{\left (a -b \right )^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left ({\mathrm e}^{2 d x +2 c} a +b \,{\mathrm e}^{2 d x +2 c}-a +b \right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right ) a^{2}}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(398\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (127) = 254\).
Time = 0.28 (sec) , antiderivative size = 1431, normalized size of antiderivative = 11.09 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=\text {Exception raised: TypeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (127) = 254\).
Time = 0.20 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d} - \frac {2 \, {\left (3 \, a^{2} b^{2} + 3 \, a b^{3} - {\left (3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} - 2 \, {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \]
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Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=-\frac {\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {d x + c}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {2 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )} - \frac {3 \, {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )}^{2} {\left (a + b\right )}^{2} {\left (a - b\right )}^{3}}}{d} \]
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Time = 2.07 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(a+b \coth (c+d x))^3} \, dx=\frac {x}{{\left (a-b\right )}^3}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (3\,a^2\,b+b^3\right )}{d\,a^6-3\,d\,a^4\,b^2+3\,d\,a^2\,b^4-d\,b^6}+\frac {2\,b^3}{d\,{\left (a+b\right )}^3\,\left (a-b\right )\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2-2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\,\left (a-b\right )\right )}-\frac {2\,\left (3\,a\,b^2-b^3\right )}{d\,{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (b-a+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\right )} \]
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