Optimal. Leaf size=89 \[ \frac {3 \text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855}
\begin {gather*} \frac {3 \text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {\tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \text {sech}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \text {sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac {3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 0.84 \begin {gather*} \frac {3 \text {ArcTan}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )+3 \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )+2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.45, size = 748, normalized size = 8.40
method | result | size |
risch | \(\frac {c^{b} \left (x^{n}\right )^{b} \left (3 \left (x^{n}\right )^{6 b} c^{6 b} {\mathrm e}^{7 a} {\mathrm e}^{-\frac {7 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {7 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {7 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {7 i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}+11 \left (x^{n}\right )^{4 b} c^{4 b} {\mathrm e}^{5 a} {\mathrm e}^{-\frac {5 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {5 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {5 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {5 i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}-11 \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {3 i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {3 i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}-3 \,{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}\right )}{4 b n \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi } {\mathrm e}^{i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi } {\mathrm e}^{-i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }+1\right )^{4}}+\frac {3 i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}+i\right )}{8 b n}-\frac {3 i \ln \left (c^{b} \left (x^{n}\right )^{b} {\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi }{2}} {\mathrm e}^{\frac {i b \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}} {\mathrm e}^{-\frac {i b \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \pi }{2}}-i\right )}{8 b n}\) | \(748\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1326 vs.
\(2 (83) = 166\).
time = 0.40, size = 1326, normalized size = 14.90 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 152, normalized size = 1.71 \begin {gather*} \frac {1}{4} \, c^{5 \, b} {\left (\frac {3 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-5 \, a\right )}}{b c^{4 \, b} c^{b} n} + \frac {{\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} + 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.35, size = 314, normalized size = 3.53 \begin {gather*} \frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}+\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n+\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}+\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n+\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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