Optimal. Leaf size=69 \[ \frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right );\frac {1}{2} \left (4+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]
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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} \[ \frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right );\frac {1}{2} \left (4+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]
Antiderivative was successfully verified.
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Rule 263
Rule 364
Rule 5545
Rule 5547
Rubi steps
\begin {align*} \int \text {sech}^2\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}^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-2 b+\frac {1}{n}}}{\left (1+e^{-2 a} x^{-2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+2 b+\frac {1}{n}}}{\left (e^{-2 a}+x^{2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right );\frac {1}{2} \left (4+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+2 b n}\\ \end {align*}
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Mathematica [A] time = 5.55, size = 126, normalized size = 1.83 \[ \frac {x \left (-\frac {e^{2 a} \left (c x^n\right )^{2 b} \, _2F_1\left (1,1+\frac {1}{2 b n};2+\frac {1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1}+\, _2F_1\left (1,\frac {1}{2 b n};1+\frac {1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )+\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{2}, x\right ) \]
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 \[ \int \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.69, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )^{2}\, dx \]
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 \[ -\frac {2 \, x}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + 4 \, \int \frac {1}{2 \, {\left (b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \operatorname {sech}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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