Integrand size = 23, antiderivative size = 198 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {(a+7 b) x}{2 (a+b)^4}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \]
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Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3756, 425, 541, 536, 212, 211} \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {b (a-3 b) (4 a+b) \tanh (c+d x)}{8 a^2 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^4}-\frac {b (2 a-b) \tanh (c+d x)}{4 a d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x (a+7 b)}{2 (a+b)^4} \]
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 541
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {a+2 b+5 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{2 (a+b) d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-2 \left (2 a^2+8 a b+3 b^2\right )-6 (2 a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 a (a+b)^2 d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {2 \left (4 a^3+24 a^2 b+11 a b^2+3 b^3\right )+2 (a-3 b) b (4 a+b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a^2 (a+b)^3 d} \\ & = \frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {(a+7 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (b^2 \left (35 a^2+14 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^4 d} \\ & = \frac {(a+7 b) x}{2 (a+b)^4}+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(2 a-b) b \tanh (c+d x)}{4 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a-3 b) b (4 a+b) \tanh (c+d x)}{8 a^2 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}
Time = 2.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {4 (a+7 b) (c+d x)+\frac {b^{3/2} \left (35 a^2+14 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2}}+2 (a+b) \sinh (2 (c+d x))+\frac {4 b^3 (a+b) \sinh (2 (c+d x))}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {b^2 (a+b) (13 a+3 b) \sinh (2 (c+d x))}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{8 (a+b)^4 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(180)=360\).
Time = 27.89 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.53
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\) | \(501\) |
default | \(\frac {-\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +7 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a +b \right )^{4}}-\frac {2 b^{2} \left (\frac {-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (39 a^{3}+98 a^{2} b +71 a \,b^{2}+12 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {\left (13 a^{2}+18 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (35 a^{2}+14 a b +3 b^{2}\right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{4}}}{d}\) | \(501\) |
risch | \(\frac {x a}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {7 x b}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {b^{2} \left (13 a^{3} {\mathrm e}^{6 d x +6 c}-a^{2} b \,{\mathrm e}^{6 d x +6 c}-17 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 \,{\mathrm e}^{6 d x +6 c} b^{3}+39 a^{3} {\mathrm e}^{4 d x +4 c}-17 a^{2} b \,{\mathrm e}^{4 d x +4 c}+33 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 \,{\mathrm e}^{4 d x +4 c} b^{3}+39 a^{3} {\mathrm e}^{2 d x +2 c}+13 a^{2} b \,{\mathrm e}^{2 d x +2 c}-35 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-9 \,{\mathrm e}^{2 d x +2 c} b^{3}+13 a^{3}+29 a^{2} b +19 a \,b^{2}+3 b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d \,a^{2}}+\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}+\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}+\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}-\frac {35 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a \left (a +b \right )^{4} d}-\frac {7 \sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{8 a^{2} \left (a +b \right )^{4} d}-\frac {3 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{3} \left (a +b \right )^{4} d}\) | \(728\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6780 vs. \(2 (180) = 360\).
Time = 0.46 (sec) , antiderivative size = 13887, normalized size of antiderivative = 70.14 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (180) = 360\).
Time = 0.57 (sec) , antiderivative size = 1806, normalized size of antiderivative = 9.12 \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )^{2}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\cosh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]
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