Optimal. Leaf size=18 \[ \frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3767, 8} \[ \frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \frac {\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {i \operatorname {Subst}\left (\int 1 \, dx,x,-i \tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 18, normalized size = 1.00 \[ \frac {\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 70, normalized size = 3.89 \[ -\frac {2}{b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 28, normalized size = 1.56 \[ -\frac {2}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 19, normalized size = 1.06 \[ \frac {\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 28, normalized size = 1.56 \[ -\frac {2}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 24, normalized size = 1.33 \[ -\frac {2}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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