Optimal. Leaf size=70 \[ \frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac {3 b+\frac {1}{n}}{2 b};\frac {1}{2} \left (5+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5545, 5547, 263, 364} \[ \frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac {3 b+\frac {1}{n}}{2 b};\frac {1}{2} \left (5+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1} \]
Antiderivative was successfully verified.
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Rule 263
Rule 364
Rule 5545
Rule 5547
Rubi steps
\begin {align*} \int \text {sech}^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \text {sech}^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (8 e^{-3 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {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 )}{n}\\ &=\frac {\left (8 e^{-3 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {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 )}{n}\\ &=\frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\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*}
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Mathematica [A] time = 0.90, size = 101, normalized size = 1.44 \[ \frac {x \left (2 e^a (b n-1) \left (c x^n\right )^b \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {1}{b n}\right );\frac {1}{2} \left (3+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )+\left (b n \tanh \left (a+b \log \left (c x^n\right )\right )+1\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )\right )}{2 b^2 n^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.00, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}\, dx \]
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 \[ 8 \, {\left (b^{2} c^{b} n^{2} - c^{b}\right )} \int \frac {e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \, {\left (b^{2} c^{2 \, b} n^{2} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b^{2} n^{2}\right )}}\,{d x} + \frac {{\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - {\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b^{2} c^{4 \, b} n^{2} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b^{2} c^{2 \, b} n^{2} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b^{2} n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int \operatorname {sech}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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