\(\int \text {sech}(a+b \log (c x^n)) \, dx\) [182]

Optimal result

Integrand size = 11, antiderivative size = 63 \[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+b n} \]

[Out]

2*exp(a)*x*(c*x^n)^b*hypergeom([1, 1/2*(b+1/n)/b],[3/2+1/2/b/n],-exp(2*a)*(c*x^n)^(2*b))/(b*n+1)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5664, 5666, 269, 371} \[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]

[In]

Int[Sech[a + b*Log[c*x^n]],x]

[Out]

(2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))])/(1 + b
*n)

Rule 269

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 5666

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[2^p/E^(a*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}, x] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \text {sech}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1-b+\frac {1}{n}}}{1+e^{-2 a} x^{-2 b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+b+\frac {1}{n}}}{e^{-2 a}+x^{2 b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {1}{b n}\right ),\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+b n} \]

[In]

Integrate[Sech[a + b*Log[c*x^n]],x]

[Out]

(2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (1 + 1/(b*n))/2, (3 + 1/(b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))])/(1 + b*n)

Maple [F]

\[\int \operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )d x\]

[In]

int(sech(a+b*ln(c*x^n)),x)

[Out]

int(sech(a+b*ln(c*x^n)),x)

Fricas [F]

\[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]

[In]

integrate(sech(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

integral(sech(b*log(c*x^n) + a), x)

Sympy [F]

\[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]

[In]

integrate(sech(a+b*ln(c*x**n)),x)

[Out]

Integral(sech(a + b*log(c*x**n)), x)

Maxima [F]

\[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]

[In]

integrate(sech(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

integrate(sech(b*log(c*x^n) + a), x)

Giac [F]

\[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]

[In]

integrate(sech(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

integrate(sech(b*log(c*x^n) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \text {sech}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]

[In]

int(1/cosh(a + b*log(c*x^n)),x)

[Out]

int(1/cosh(a + b*log(c*x^n)), x)