Integrand size = 23, antiderivative size = 185 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {(a-5 b) x}{2 (a+b)^4}-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {3 b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(11 a-b) b \tanh (c+d x)}{8 a (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \]
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Time = 0.19 (sec) , antiderivative size = 185, 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 = {3744, 482, 541, 536, 212, 211} \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} d (a+b)^4}-\frac {b (11 a-b) \tanh (c+d x)}{8 a d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 b \tanh (c+d x)}{4 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {x (a-5 b)}{2 (a+b)^4} \]
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Rule 211
Rule 212
Rule 482
Rule 536
Rule 541
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{\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-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 {3 b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-2 a (2 a-b)+18 a 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 {3 b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(11 a-b) b \tanh (c+d x)}{8 a (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 a \left (4 a^2-9 a b-b^2\right )-2 a (11 a-b) 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 {3 b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(11 a-b) b \tanh (c+d x)}{8 a (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-5 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 \left (15 a^2-10 a b-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a (a+b)^4 d} \\ & = -\frac {(a-5 b) x}{2 (a+b)^4}-\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (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 {3 b \tanh (c+d x)}{4 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(11 a-b) b \tanh (c+d x)}{8 a (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}
Time = 2.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.85 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {-4 (a-5 b) (c+d x)+\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+2 (a+b) \sinh (2 (c+d x))-\frac {4 b^2 (a+b) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}-\frac {(9 a-b) b (a+b) \sinh (2 (c+d x))}{a (a-b+(a+b) \cosh (2 (c+d x)))}}{8 (a+b)^4 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(167)=334\).
Time = 15.26 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.64
method | result | size |
derivativedivides | \(\frac {\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 -5 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 \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (27 a^{3}+58 a^{2} b +27 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a}-\frac {\left (27 a^{3}+58 a^{2} b +27 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a}+\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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 (15 a^{2}-10 a b -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}\right )}{\left (a +b \right )^{4}}-\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 +5 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}}{d}\) | \(488\) |
default | \(\frac {\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 -5 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 \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (27 a^{3}+58 a^{2} b +27 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a}-\frac {\left (27 a^{3}+58 a^{2} b +27 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a}+\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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 (15 a^{2}-10 a b -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}\right )}{\left (a +b \right )^{4}}-\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 +5 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )^{4}}}{d}\) | \(488\) |
risch | \(-\frac {x a}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {5 x b}{2 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {{\mathrm e}^{2 d x +2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {b \left (9 a^{3} {\mathrm e}^{6 d x +6 c}-5 a^{2} b \,{\mathrm e}^{6 d x +6 c}-13 a \,b^{2} {\mathrm e}^{6 d x +6 c}+{\mathrm e}^{6 d x +6 c} b^{3}+27 a^{3} {\mathrm e}^{4 d x +4 c}-21 a^{2} b \,{\mathrm e}^{4 d x +4 c}+29 a \,b^{2} {\mathrm e}^{4 d x +4 c}-3 \,{\mathrm e}^{4 d x +4 c} b^{3}+27 a^{3} {\mathrm e}^{2 d x +2 c}+a^{2} b \,{\mathrm e}^{2 d x +2 c}-23 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+3 \,{\mathrm e}^{2 d x +2 c} b^{3}+9 a^{3}+17 a^{2} b +7 a \,b^{2}-b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) a d \left (a +b \right )}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 \left (a +b \right )^{4} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{8 a \left (a +b \right )^{4} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 \left (a +b \right )^{4} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{8 a \left (a +b \right )^{4} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} d}\) | \(712\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6319 vs. \(2 (167) = 334\).
Time = 0.45 (sec) , antiderivative size = 12965, normalized size of antiderivative = 70.08 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (167) = 334\).
Time = 0.59 (sec) , antiderivative size = 1806, normalized size of antiderivative = 9.76 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]
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