Optimal. Leaf size=69 \[ \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+4 b n} \]
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Rubi [A]
time = 0.05, 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 = {5664, 5666,
269, 371} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 269
Rule 371
Rule 5664
Rule 5666
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 ) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(69)=138\).
time = 9.34, size = 192, normalized size = 2.78 \begin {gather*} \frac {x \left (-2 e^{2 a} (-1+2 b n) \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 )+\left (-2+8 b^2 n^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 (2 b n+\left (-1+8 b^2 n^2+\left (-1+4 b^2 n^2\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )\right )\right )}{12 b^3 n^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.68, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \operatorname {sech}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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