Integrand size = 21, antiderivative size = 95 \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {\log (\tanh (c+d x))}{a^2 d}-\frac {b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac {b}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3751, 457, 84} \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)^2}+\frac {\log (\tanh (c+d x))}{a^2 d}+\frac {b}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rule 84
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{(1-x) x (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}+\frac {1}{a^2 x}-\frac {b^2}{a (a+b) (a+b x)^2}-\frac {b^2 (2 a+b)}{a^2 (a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = \frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {\log (\tanh (c+d x))}{a^2 d}-\frac {b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac {b}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}
Time = 2.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {2 \log (\cosh (c+d x))}{(a+b)^2}+\frac {2 \log (\tanh (c+d x))+\frac {b \left (-\left ((2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )\right )+\frac {a (a+b)}{a+b \tanh ^2(c+d x)}\right )}{(a+b)^2}}{a^2}}{2 d} \]
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Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (d x +c \right )\right )}{a^{2}}+\frac {b^{2} \left (\frac {\left (2 a +b \right ) \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{b}-\frac {a \left (a +b \right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}\right )}{2 \left (a +b \right )^{2} a^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{2}}}{d}\) | \(110\) |
default | \(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (d x +c \right )\right )}{a^{2}}+\frac {b^{2} \left (\frac {\left (2 a +b \right ) \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{b}-\frac {a \left (a +b \right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}\right )}{2 \left (a +b \right )^{2} a^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{2}}}{d}\) | \(110\) |
parallelrisch | \(\frac {-2 \left (a +b \tanh \left (d x +c \right )^{2}\right ) b \left (a +\frac {b}{2}\right ) \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )+\left (-2 \tanh \left (d x +c \right )^{2} a^{2} b -2 a^{3}\right ) \ln \left (1-\tanh \left (d x +c \right )\right )+2 \left (a +b \right )^{2} \left (a +b \tanh \left (d x +c \right )^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )+\left (-2 a^{2} b d x -a \,b^{2}-b^{3}\right ) \tanh \left (d x +c \right )^{2}-2 a^{3} d x}{2 \left (a +b \tanh \left (d x +c \right )^{2}\right ) d \left (a +b \right )^{2} a^{2}}\) | \(156\) |
risch | \(\frac {x}{a^{2}+2 a b +b^{2}}-\frac {2 x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {4 b x}{a \left (a^{2}+2 a b +b^{2}\right )}+\frac {4 b c}{d a \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} x}{a^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} c}{d \,a^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c}}{a d \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}-\frac {b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{d a \left (a^{2}+2 a b +b^{2}\right )}-\frac {b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \,a^{2} \left (a^{2}+2 a b +b^{2}\right )}\) | \(329\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (91) = 182\).
Time = 0.36 (sec) , antiderivative size = 1148, normalized size of antiderivative = 12.08 \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\coth {\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (91) = 182\).
Time = 0.22 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.47 \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + 2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {{\left (2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (91) = 182\).
Time = 0.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.05 \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (2 \, a b + b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{4} + 2 \, a^{3} b + a^{2} b^{2}} + \frac {2 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} - \frac {4 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} {\left (a + b\right )}^{2} a} - \frac {2 \, \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}}}{2 \, d} \]
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Timed out. \[ \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]
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