Integrand size = 23, antiderivative size = 178 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {x}{(a+b)^3}-\frac {b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a+b)^3 d}-\frac {\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]
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Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3751, 483, 593, 597, 536, 212, 211} \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d (a+b)^3}-\frac {\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 d (a+b)^2}+\frac {b \coth (c+d x)}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \]
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Rule 211
Rule 212
Rule 483
Rule 536
Rule 593
Rule 597
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 a-5 b+5 b x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b) d} \\ & = \frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2+27 a b+15 b^2-3 b (9 a+5 b) x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d} \\ & = -\frac {\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-8 a^3+8 a^2 b+27 a b^2+15 b^3-b \left (8 a^2+27 a b+15 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^2 d} \\ & = -\frac {\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^3 d}-\frac {\left (b^2 \left (35 a^2+42 a b+15 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^3 d} \\ & = \frac {x}{(a+b)^3}-\frac {b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a+b)^3 d}-\frac {\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}
Time = 6.80 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {-\frac {8 (c+d x)}{(a+b)^3}+\frac {b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} (a+b)^3}+\frac {8 \coth (c+d x)}{a^3}+\frac {4 b^3 \sinh (2 (c+d x))}{a^2 (a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {b^2 (13 a+7 b) \sinh (2 (c+d x))}{a^3 (a+b)^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{8 d} \]
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Time = 0.42 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}+\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{2} b +\frac {9}{4} a \,b^{2}+\frac {7}{8} b^{3}\right ) \tanh \left (d x +c \right )^{3}+\frac {a \left (13 a^{2}+22 a b +9 b^{2}\right ) \tanh \left (d x +c \right )}{8}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (35 a^{2}+42 a b +15 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{3} a^{3}}+\frac {1}{a^{3} \tanh \left (d x +c \right )}+\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(167\) |
default | \(-\frac {-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}+\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{2} b +\frac {9}{4} a \,b^{2}+\frac {7}{8} b^{3}\right ) \tanh \left (d x +c \right )^{3}+\frac {a \left (13 a^{2}+22 a b +9 b^{2}\right ) \tanh \left (d x +c \right )}{8}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (35 a^{2}+42 a b +15 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a +b \right )^{3} a^{3}}+\frac {1}{a^{3} \tanh \left (d x +c \right )}+\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(167\) |
risch | \(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {-60 b^{5} {\mathrm e}^{6 d x +6 c}+90 b^{5} {\mathrm e}^{4 d x +4 c}+8 a^{5}+15 b^{5}+40 a^{4} b +93 a^{3} b^{2}+113 a^{2} b^{3}+67 a \,b^{4}-60 b^{5} {\mathrm e}^{2 d x +2 c}+66 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}-138 a \,b^{4} {\mathrm e}^{6 d x +6 c}+172 a \,b^{4} {\mathrm e}^{4 d x +4 c}-158 a \,b^{4} {\mathrm e}^{2 d x +2 c}+77 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}+57 a \,b^{4} {\mathrm e}^{8 d x +8 c}+15 b^{5} {\mathrm e}^{8 d x +8 c}+32 a^{5} {\mathrm e}^{6 d x +6 c}+8 a^{5} {\mathrm e}^{8 d x +8 c}+32 \,{\mathrm e}^{2 d x +2 c} a^{5}+48 \,{\mathrm e}^{4 d x +4 c} a^{5}-56 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}+40 \,{\mathrm e}^{8 d x +8 c} a^{4} b +67 \,{\mathrm e}^{8 d x +8 c} a^{3} b^{2}+96 \,{\mathrm e}^{2 d x +2 c} a^{4} b +90 \,{\mathrm e}^{2 d x +2 c} a^{3} b^{2}-72 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{3}+112 \,{\mathrm e}^{4 d x +4 c} a^{4} b +96 \,{\mathrm e}^{4 d x +4 c} a^{3} b^{2}+96 b \,a^{4} {\mathrm e}^{6 d x +6 c}+38 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}}{4 \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 )^{2} \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) a^{3} d \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{16 a^{2} \left (a +b \right )^{3} d}+\frac {21 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b^{2}}{8 a^{3} \left (a +b \right )^{3} d}+\frac {15 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{4} \left (a +b \right )^{3} d}-\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{16 a^{2} \left (a +b \right )^{3} d}-\frac {21 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b^{2}}{8 a^{3} \left (a +b \right )^{3} d}-\frac {15 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{4} \left (a +b \right )^{3} d}\) | \(833\) |
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Leaf count of result is larger than twice the leaf count of optimal. 5772 vs. \(2 (162) = 324\).
Time = 0.44 (sec) , antiderivative size = 11865, normalized size of antiderivative = 66.66 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1944 vs. \(2 (162) = 324\).
Time = 0.65 (sec) , antiderivative size = 1944, normalized size of antiderivative = 10.92 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (162) = 324\).
Time = 0.46 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.46 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {{\left (35 \, a^{2} b^{2} + 42 \, a b^{3} + 15 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {a b}} - \frac {8 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left (13 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 17 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 7 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 5 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 25 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 21 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b^{2} + 33 \, a^{2} b^{3} + 27 \, a b^{4} + 7 \, b^{5}\right )}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} + \frac {16}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \]
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Timed out. \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]
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