Optimal. Leaf size=69 \[ \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{4 b n+1} \]
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Rubi [A] time = 0.07, antiderivative size = 69, 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} \[ \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{4 b n+1} \]
Antiderivative was successfully verified.
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Rule 263
Rule 364
Rule 5545
Rule 5547
Rubi steps
\begin {align*} \int \text {sech}^4\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}^4(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (16 e^{-4 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-4 b+\frac {1}{n}}}{\left (1+e^{-2 a} x^{-2 b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (16 e^{-4 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+4 b+\frac {1}{n}}}{\left (e^{-2 a}+x^{2 b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+4 b n}\\ \end {align*}
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Mathematica [B] time = 13.70, size = 192, normalized size = 2.78 \[ \frac {x \left (\left (8 b^2 n^2-2\right ) \, _2F_1\left (1,\frac {1}{2 b n};1+\frac {1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )+\text {sech}^2\left (a+b \log \left (c x^n\right )\right ) \left (\tanh \left (a+b \log \left (c x^n\right )\right ) \left (\left (4 b^2 n^2-1\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+8 b^2 n^2-1\right )+2 b n\right )-2 e^{2 a} (2 b n-1) \left (c x^n\right )^{2 b} \, _2F_1\left (1,1+\frac {1}{2 b n};2+\frac {1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )\right )}{12 b^3 n^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{4}, x\right ) \]
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 \[ \int \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.83, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )^{4}\, dx \]
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 \[ 16 \, {\left (4 \, b^{2} n^{2} - 1\right )} \int \frac {1}{48 \, {\left (b^{3} c^{2 \, b} n^{3} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b^{3} n^{3}\right )}}\,{d x} + \frac {{\left (2 \, b c^{4 \, b} n + c^{4 \, b}\right )} x e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, {\left (6 \, b^{2} c^{2 \, b} n^{2} - b c^{2 \, b} n - c^{2 \, b}\right )} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - {\left (4 \, b^{2} n^{2} - 1\right )} x}{3 \, {\left (b^{3} c^{6 \, b} n^{3} e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 3 \, b^{3} c^{4 \, b} n^{3} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b^{3} c^{2 \, b} n^{3} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b^{3} n^{3}\right )}} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]
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 \[ \int \operatorname {sech}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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