Integrand size = 18, antiderivative size = 102 \[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-a} x \left (c x^n\right )^{-2/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}} \]
1/2*x*cosh(a+2*ln(c*x^n)/n)^(1/2)-1/2*x*arccsch(exp(a)*(c*x^n)^(2/n))*cosh (a+2*ln(c*x^n)/n)^(1/2)/exp(a)/((c*x^n)^(2/n))/(1+1/exp(2*a)/((c*x^n)^(4/n )))^(1/2)
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.73 \[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{2} x \left (1-\frac {\text {arctanh}\left (\sqrt {1+e^{2 a} \left (c x^n\right )^{4/n}}\right )}{\sqrt {1+e^{2 a} \left (c x^n\right )^{4/n}}}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
(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
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6052, 6060, 868, 773, 247, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx\) |
\(\Big \downarrow \) 6052 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 6060 |
\(\displaystyle \frac {x \left (c x^n\right )^{-2/n} \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \left (c x^n\right )^{\frac {2}{n}-1} \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}d\left (c x^n\right )}{n \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}\) |
\(\Big \downarrow \) 868 |
\(\displaystyle \frac {x \left (c x^n\right )^{-2/n} \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \sqrt {\frac {e^{-2 a} x^{-2 n}}{c^2}+1}d\left (c x^n\right )^{2/n}}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}\) |
\(\Big \downarrow \) 773 |
\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \int \frac {x^{-2 n} \sqrt {c^2 e^{-2 a} x^{2 n}+1}}{c^2}d\frac {x^{-n}}{c}}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \left (e^{-2 a} \int \frac {1}{\sqrt {c^2 e^{-2 a} x^{2 n}+1}}d\frac {x^{-n}}{c}-\frac {x^{-n} \sqrt {e^{-2 a} c^2 x^{2 n}+1}}{c}\right )}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {x \left (c x^n\right )^{-2/n} \left (e^{-a} \text {arcsinh}\left (\frac {e^{-a} x^{-n}}{c}\right )-\frac {x^{-n} \sqrt {e^{-2 a} c^2 x^{2 n}+1}}{c}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}}\) |
-1/2*(x*(-(Sqrt[1 + (c^2*x^(2*n))/E^(2*a)]/(c*x^n)) + ArcSinh[1/(c*E^a*x^n )]/E^a)*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/((c*x^n)^(2/n)*Sqrt[1 + 1/(E^(2* a)*(c*x^n)^(4/n))])
3.3.59.3.1 Defintions of rubi rules used
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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] && !IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]
Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S imp[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] )
Int[Cosh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[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] /; FreeQ[ {a, b, d, e, m, p}, x] && !IntegerQ[p]
\[\int \sqrt {\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}d x\]
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.38 \[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\frac {1}{8} \, {\left (4 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )} + \sqrt {2} e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )} \log \left (\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 2 \, \sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} + 2}{x^{4}}\right )\right )} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{n}\right )} \]
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)
\[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int \sqrt {\cosh {\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \]
\[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int { \sqrt {\cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )} \,d x } \]
Timed out. \[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx=\int \sqrt {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )} \,d x \]