Integrand size = 23, antiderivative size = 156 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\arctan (\sinh (c+d x))}{b^3 d}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]
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Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3757, 425, 541, 536, 209, 211} \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left ((a+b) \sinh ^2(c+d x)+a\right )^2}-\frac {\arctan (\sinh (c+d x))}{b^3 d} \]
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Rule 209
Rule 211
Rule 425
Rule 536
Rule 541
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a-3 b-3 (a+b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a b d} \\ & = \frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {4 a^2-a b+3 b^2-(4 a-3 b) (a+b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 a^2 b^2 d} \\ & = \frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b^3 d}+\frac {\left ((a+b) \left (8 a^2-4 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 b^3 d} \\ & = -\frac {\arctan (\sinh (c+d x))}{b^3 d}+\frac {\sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}+\frac {(a+b) \sinh (c+d x)}{4 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {(4 a-3 b) (a+b) \sinh (c+d x)}{8 a^2 b^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.69 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.03 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {2 \sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{5/2}}+\frac {2 \left (8 a^3+4 a^2 b-a b^2+3 b^3\right ) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b}}+64 \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {i \sqrt {a+b} \left (8 a^2-4 a b+3 b^2\right ) \log (a-b+(a+b) \cosh (2 (c+d x)))}{a^{5/2}}-\frac {i \left (8 a^3+4 a^2 b-a b^2+3 b^3\right ) \log (a-b+(a+b) \cosh (2 (c+d x)))}{a^{5/2} \sqrt {a+b}}-\frac {32 b^2 (a+b) \sinh (c+d x)}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {8 b \left (4 a^2+a b-3 b^2\right ) \sinh (c+d x)}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}}{32 b^3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(142)=284\).
Time = 0.21 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.49
\[\frac {\frac {\frac {2 \left (\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}+\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}-\frac {\left (4 a^{3}+23 a^{2} b +7 a \,b^{2}-12 b^{3}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}-\frac {b \left (4 a^{2}-a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (8 a^{3}+4 a^{2} b -a \,b^{2}+3 b^{3}\right ) \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{4 a}}{b^{3}}-\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\]
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Leaf count of result is larger than twice the leaf count of optimal. 4457 vs. \(2 (142) = 284\).
Time = 0.38 (sec) , antiderivative size = 8070, normalized size of antiderivative = 51.73 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{7}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{7}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^7\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]
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