3.259 \(\int \sqrt{\cosh (a+\frac{2 \log (c x^n)}{n})} \, dx\)

Optimal. Leaf size=102 \[ \frac{1}{2} x \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}-\frac{e^{-a} x \left (c x^n\right )^{-2/n} \text{csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt{e^{-2 a} \left (c x^n\right )^{-4/n}+1}} \]

[Out]

(x*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/2 - (x*ArcCsch[E^a*(c*x^n)^(2/n)]*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/(2*E^
a*(c*x^n)^(2/n)*Sqrt[1 + 1/(E^(2*a)*(c*x^n)^(4/n))])

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Rubi [A]  time = 0.0823893, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5526, 5534, 345, 242, 277, 215} \[ \frac{1}{2} x \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}-\frac{e^{-a} x \left (c x^n\right )^{-2/n} \text{csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt{e^{-2 a} \left (c x^n\right )^{-4/n}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cosh[a + (2*Log[c*x^n])/n]],x]

[Out]

(x*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/2 - (x*ArcCsch[E^a*(c*x^n)^(2/n)]*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/(2*E^
a*(c*x^n)^(2/n)*Sqrt[1 + 1/(E^(2*a)*(c*x^n)^(4/n))])

Rule 5526

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Cosh[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 5534

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

Rule 345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sqrt{\cosh \left (a+\frac{2 \log (x)}{n}\right )} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-2/n} \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{2}{n}} \sqrt{1+e^{-2 a} x^{-4/n}} \, dx,x,c x^n\right )}{n \sqrt{1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac{\left (x \left (c x^n\right )^{-2/n} \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\right ) \operatorname{Subst}\left (\int \sqrt{1+\frac{e^{-2 a}}{x^2}} \, dx,x,\left (c x^n\right )^{2/n}\right )}{2 \sqrt{1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=-\frac{\left (x \left (c x^n\right )^{-2/n} \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+e^{-2 a} x^2}}{x^2} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{2 \sqrt{1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac{1}{2} x \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}-\frac{\left (e^{-2 a} x \left (c x^n\right )^{-2/n} \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+e^{-2 a} x^2}} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{2 \sqrt{1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac{1}{2} x \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}-\frac{e^{-a} x \left (c x^n\right )^{-2/n} \sinh ^{-1}\left (e^{-a} \left (c x^n\right )^{-2/n}\right ) \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt{1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ \end{align*}

Mathematica [A]  time = 0.31763, size = 74, normalized size = 0.73 \[ \frac{1}{2} x \left (1-\frac{\tanh ^{-1}\left (\sqrt{e^{2 a} \left (c x^n\right )^{4/n}+1}\right )}{\sqrt{e^{2 a} \left (c x^n\right )^{4/n}+1}}\right ) \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[Cosh[a + (2*Log[c*x^n])/n]],x]

[Out]

(x*(1 - ArcTanh[Sqrt[1 + E^(2*a)*(c*x^n)^(4/n)]]/Sqrt[1 + E^(2*a)*(c*x^n)^(4/n)])*Sqrt[Cosh[a + (2*Log[c*x^n])
/n]])/2

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Maple [F]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cosh \left ( a+2\,{\frac{\ln \left ( c{x}^{n} \right ) }{n}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+2*ln(c*x^n)/n)^(1/2),x)

[Out]

int(cosh(a+2*ln(c*x^n)/n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cosh \left (a + \frac{2 \, \log \left (c x^{n}\right )}{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(cosh(a + 2*log(c*x^n)/n)), x)

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Fricas [A]  time = 1.79661, size = 360, normalized size = 3.53 \begin{align*} \frac{1}{8} \,{\left (4 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (\frac{a n + 2 \, \log \left (c\right )}{2 \, n}\right )} + \sqrt{2} e^{\left (\frac{a n + 2 \, \log \left (c\right )}{2 \, n}\right )} \log \left (\frac{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 2 \, \sqrt{2} \sqrt{\frac{1}{2}} x \sqrt{\frac{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} + 2}{x^{4}}\right )\right )} e^{\left (-\frac{a n + 2 \, \log \left (c\right )}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="fricas")

[Out]

1/8*(4*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) + 1)/x^2)*e^(1/2*(a*n + 2*log(c))/n) + sqrt(2)*e^(1/2*(a
*n + 2*log(c))/n)*log((x^4*e^(2*(a*n + 2*log(c))/n) - 2*sqrt(2)*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n)
+ 1)/x^2) + 2)/x^4))*e^(-(a*n + 2*log(c))/n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cosh{\left (a + \frac{2 \log{\left (c x^{n} \right )}}{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+2*ln(c*x**n)/n)**(1/2),x)

[Out]

Integral(sqrt(cosh(a + 2*log(c*x**n)/n)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+2*log(c*x^n)/n)^(1/2),x, algorithm="giac")

[Out]

Timed out