Integrand size = 44, antiderivative size = 40 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {sech}\left (a+b \log \left (c x^n\right )\right )+b n x \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.48, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5664, 5666, 269, 371} \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {16 e^{3 a} b^2 n^2 x \left (c x^n\right )^{3 b} \operatorname {Hypergeometric2F1}\left (3,\frac {3 b+\frac {1}{n}}{2 b},\frac {1}{2} \left (5+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1}+2 e^a x (1-b n) \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right ) \]
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Rule 269
Rule 371
Rule 5664
Rule 5666
Rubi steps \begin{align*} \text {integral}& = \left (2 b^2 n^2\right ) \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx+\left (1-b^2 n^2\right ) \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \left (2 b^2 n x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \text {sech}^3(a+b \log (x)) \, dx,x,c x^n\right )+\frac {\left (\left (1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \text {sech}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \left (16 b^2 e^{-3 a} n x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1-3 b+\frac {1}{n}}}{\left (1+e^{-2 a} x^{-2 b}\right )^3} \, dx,x,c x^n\right )+\frac {\left (2 e^{-a} \left (1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1-b+\frac {1}{n}}}{1+e^{-2 a} x^{-2 b}} \, dx,x,c x^n\right )}{n} \\ & = \left (16 b^2 e^{-3 a} n x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+3 b+\frac {1}{n}}}{\left (e^{-2 a}+x^{2 b}\right )^3} \, dx,x,c x^n\right )+\frac {\left (2 e^{-a} \left (1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+b+\frac {1}{n}}}{e^{-2 a}+x^{2 b}} \, dx,x,c x^n\right )}{n} \\ & = 2 e^a (1-b n) x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )+\frac {16 b^2 e^{3 a} n^2 x \left (c x^n\right )^{3 b} \operatorname {Hypergeometric2F1}\left (3,\frac {3 b+\frac {1}{n}}{2 b},\frac {1}{2} \left (5+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+3 b n} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {sech}\left (a+b \log \left (c x^n\right )\right ) \left (1+b n \tanh \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Time = 33.73 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.25
method | result | size |
parallelrisch | \(\frac {2 x \left (b n \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{\cosh \left (4 b \ln \left (\sqrt {c \,x^{n}}\right )+2 a \right )+1}\) | \(50\) |
risch | \(\frac {2 c^{b} \left (x^{n}\right )^{b} x \left (n b \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}-{\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} b n +\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}+{\mathrm e}^{a} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}}\right )}{{\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{2}}\) | \(509\) |
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (40) = 80\).
Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.72 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, {\left ({\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (b n + 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (b n - 1\right )} x\right )}}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
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\[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (2 b^{2} n^{2} \operatorname {sech}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} - b^{2} n^{2} + 1\right ) \operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (40) = 80\).
Time = 0.36 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.40 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, {\left ({\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - {\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}\right )}}{c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (40) = 80\).
Time = 0.76 (sec) , antiderivative size = 215, normalized size of antiderivative = 5.38 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, b c^{3 \, b} n x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} - \frac {2 \, b c^{b} n x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} + \frac {2 \, c^{3 \, b} x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} + \frac {2 \, c^{b} x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} \]
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Time = 2.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.65 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2\,x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b\,\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{{\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}^2} \]
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