3.2.89 \(\int \text {sech}^p(a+\frac {\log (c x^n)}{n (-2+p)}) \, dx\) [189]

Optimal. Leaf size=89 \[ \frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1+e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

[Out]

1/2*exp(2*a)*(2-p)*x*(1+(c*x^n)^(2/n/(2-p))/exp(2*a))*sech(a-ln(c*x^n)/n/(2-p))^p/(1-p)/((c*x^n)^(2/n/(2-p)))

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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5664, 5668, 267} \begin {gather*} \frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}+1\right ) \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[a + Log[c*x^n]/(n*(-2 + p))]^p,x]

[Out]

(E^(2*a)*(2 - p)*x*(1 + (c*x^n)^(2/(n*(2 - p)))/E^(2*a))*Sech[a - Log[c*x^n]/(n*(2 - p))]^p)/(2*(1 - p)*(c*x^n
)^(2/(n*(2 - p))))

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5664

Int[Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5668

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[Sech[d*(a + b*Log[x])]^p*(
(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p/x^((-b)*d*p)), Int[(e*x)^m*(1/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)), x]
, x] /; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \text {sech}^p\left (a+\frac {\log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {1}{n}+\frac {p}{n (-2+p)}} \left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}-\frac {p}{n (-2+p)}} \left (1+e^{-2 a} x^{-\frac {2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1+e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end {align*}

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Mathematica [A]
time = 3.67, size = 114, normalized size = 1.28 \begin {gather*} -\frac {2^{-1+p} (-2+p) x \left (\frac {e^a \left (c x^n\right )^{\frac {1}{2 n-n p}}}{e^{2 a}+\left (c x^n\right )^{-\frac {2}{n (-2+p)}}}\right )^p \left (-1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}} \left (-1+\left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p\right )\right )}{-1+p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + Log[c*x^n]/(n*(-2 + p))]^p,x]

[Out]

-((2^(-1 + p)*(-2 + p)*x*((E^a*(c*x^n)^(2*n - n*p)^(-1))/(E^(2*a) + (c*x^n)^(-2/(n*(-2 + p)))))^p*(-1 + E^(2*a
)*(c*x^n)^(2/(n*(-2 + p)))*(-1 + (1 + 1/(E^(2*a)*(c*x^n)^(2/(n*(-2 + p)))))^p)))/(-1 + p))

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Maple [F]
time = 1.67, size = 0, normalized size = 0.00 \[\int \mathrm {sech}\left (a +\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+ln(c*x^n)/n/(-2+p))^p,x)

[Out]

int(sech(a+ln(c*x^n)/n/(-2+p))^p,x)

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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.

[In]

integrate(sech(a+log(c*x^n)/n/(-2+p))^p,x, algorithm="maxima")

[Out]

integrate(sech(a + log(c*x^n)/(n*(p - 2)))^p, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (76) = 152\).
time = 0.38, size = 474, normalized size = 5.33 \begin {gather*} \frac {{\left (p - 2\right )} x \cosh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 1}\right )\right ) \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + {\left (p - 2\right )} x \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 1}\right )\right )}{{\left (p - 1\right )} \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) - {\left (p - 1\right )} \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+log(c*x^n)/n/(-2+p))^p,x, algorithm="fricas")

[Out]

((p - 2)*x*cosh(p*log(2*(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n*log(x
) + log(c))/(n*p - 2*n)))/(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))^2 + 2*cosh((a*n*p - 2*a*n + n
*log(x) + log(c))/(n*p - 2*n))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n
*log(x) + log(c))/(n*p - 2*n))^2 + 1)))*cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + (p - 2)*x*cosh
((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))*sinh(p*log(2*(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p
- 2*n)) + sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)))/(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*
p - 2*n))^2 + 2*cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))
/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))^2 + 1))))/((p - 1)*cosh((a*n*p - 2*a*n +
n*log(x) + log(c))/(n*p - 2*n)) - (p - 1)*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {sech}^{p}{\left (a + \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+ln(c*x**n)/n/(-2+p))**p,x)

[Out]

Integral(sech(a + log(c*x**n)/(n*(p - 2)))**p, x)

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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.

[In]

integrate(sech(a+log(c*x^n)/n/(-2+p))^p,x, algorithm="giac")

[Out]

integrate(sech(a + log(c*x^n)/(n*(p - 2)))^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {1}{\mathrm {cosh}\left (a+\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(a + log(c*x^n)/(n*(p - 2))))^p,x)

[Out]

int((1/cosh(a + log(c*x^n)/(n*(p - 2))))^p, x)

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