Integrand size = 18, antiderivative size = 206 \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {1}{4} x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \]
-1/4*x*cosh(a+2*ln(c*x^n)/n)^(5/2)+5/4*x*cosh(a+2*ln(c*x^n)/n)^(5/2)/exp(2 *a)/((c*x^n)^(4/n))/(1+1/exp(2*a)/((c*x^n)^(4/n)))^2+5/12*x*cosh(a+2*ln(c* x^n)/n)^(5/2)/(1+1/exp(2*a)/((c*x^n)^(4/n)))-5/4*x*arccsch(exp(a)*(c*x^n)^ (2/n))*cosh(a+2*ln(c*x^n)/n)^(5/2)/exp(3*a)/((c*x^n)^(6/n))/(1+1/exp(2*a)/ ((c*x^n)^(4/n)))^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.41 \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{14} e^{2 a} x \left (c x^n\right )^{4/n} \left (1+e^{2 a} \left (c x^n\right )^{4/n}\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},1+e^{2 a} \left (c x^n\right )^{4/n}\right ) \]
(E^(2*a)*x*(c*x^n)^(4/n)*(1 + E^(2*a)*(c*x^n)^(4/n))*Cosh[a + (2*Log[c*x^n ])/n]^(5/2)*Hypergeometric2F1[2, 7/2, 9/2, 1 + E^(2*a)*(c*x^n)^(4/n)])/14
Time = 0.46 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6052, 6060, 876, 872, 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.
\(\displaystyle \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx\) |
\(\Big \downarrow \) 6052 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 6060 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \int \left (c x^n\right )^{\frac {6}{n}-1} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}d\left (c x^n\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 876 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \int \left (c x^n\right )^{\frac {6}{n}-1} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}d\left (c x^n\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 872 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (e^{-2 a} \int \left (c x^n\right )^{\frac {2}{n}-1} \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}d\left (c x^n\right )+\frac {1}{6} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 868 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{2} e^{-2 a} n \int \sqrt {\frac {e^{-2 a} x^{-2 n}}{c^2}+1}d\left (c x^n\right )^{2/n}+\frac {1}{6} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 773 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{6} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}-\frac {1}{2} e^{-2 a} n \int \frac {x^{-2 n} \sqrt {c^2 e^{-2 a} x^{2 n}+1}}{c^2}d\frac {x^{-n}}{c}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{6} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}-\frac {1}{2} e^{-2 a} n \left (e^{-2 a} \int \frac {1}{\sqrt {c^2 e^{-2 a} x^{2 n}+1}}d\frac {x^{-n}}{c}-\frac {x^{-n} \sqrt {e^{-2 a} c^2 x^{2 n}+1}}{c}\right )\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \left (\frac {5}{2} \left (\frac {1}{6} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{3/2}-\frac {1}{2} e^{-2 a} n \left (e^{-a} \text {arcsinh}\left (\frac {e^{-a} x^{-n}}{c}\right )-\frac {x^{-n} \sqrt {e^{-2 a} c^2 x^{2 n}+1}}{c}\right )\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{n \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}}\) |
(x*(-1/4*(n*(c*x^n)^(6/n)*(1 + 1/(E^(2*a)*(c*x^n)^(4/n)))^(5/2)) + (5*((n* (c*x^n)^(6/n)*(1 + 1/(E^(2*a)*(c*x^n)^(4/n)))^(3/2))/6 - (n*(-(Sqrt[1 + (c ^2*x^(2*n))/E^(2*a)]/(c*x^n)) + ArcSinh[1/(c*E^a*x^n)]/E^a))/(2*E^(2*a)))) /2)*Cosh[a + (2*Log[c*x^n])/n]^(5/2))/(n*(c*x^n)^(6/n)*(1 + 1/(E^(2*a)*(c* x^n)^(4/n)))^(5/2))
3.3.58.3.1 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*( (a + b*x^n)^p/(m + 1)), x] - Simp[b*n*(p/(m + 1)) Int[x^(m + n)*(a + b*x^ n)^(p - 1), x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[(m + 1)/n + p, 0] && Gt Q[p, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m, n}, x] & & IntegerQ[p + Simplify[(m + 1)/n]] && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
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]
\[\int {\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}^{\frac {5}{2}}d x\]
Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.91 \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {{\left (15 \, \sqrt {2} x^{3} e^{\left (\frac {3 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{2 \, n}\right )} \log \left (\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 2 \, \sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} + 2}{x^{4}}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (2 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 14 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 3\right )} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}\right )} e^{\left (-\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )}}{192 \, x^{3}} \]
1/192*(15*sqrt(2)*x^3*e^(3/2*(a*n + 2*log(c))/n)*log((x^4*e^(2*(a*n + 2*lo g(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) + 4*sqrt(1/2)*(2*x^8*e^(4*(a*n + 2*log(c))/n) + 14*x^4*e^(2* (a*n + 2*log(c))/n) - 3)*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) + 1)/x^2)*e^(- 1/2*(a*n + 2*log(c))/n))*e^(-2*(a*n + 2*log(c))/n)/x^3
Timed out. \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\text {Timed out} \]
\[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int { \cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {5}{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3088 vs. \(2 (192) = 384\).
Time = 135.03 (sec) , antiderivative size = 3088, normalized size of antiderivative = 14.99 \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\text {Too large to display} \]
1/48*sqrt(2)*sqrt(c^(6/n)*x^6*e^(3*a) + c^(2/n)*x^2*e^a)*c^(2/n)*x^3*e^a + 1/192*sqrt(2)*(15*c^(8/n)*e^(9/2*a)*log(sqrt(-sqrt(x^8*abs(c)^(8/n)*e^(4* a) + 2*x^4*abs(c)^(4/n)*cos(2*pi*sgn(c)/n - 2*pi/n)*e^(2*a) + 1)*cos(1/2*p i*sgn(x^4*abs(c)^(4/n)*cos(-1/2*pi*sgn(c)/n + 1/2*pi/n)^4*e^(3*a) - 6*x^4* abs(c)^(4/n)*cos(-1/2*pi*sgn(c)/n + 1/2*pi/n)^2*e^(3*a)*sin(-1/2*pi*sgn(c) /n + 1/2*pi/n)^2 + x^4*abs(c)^(4/n)*e^(3*a)*sin(-1/2*pi*sgn(c)/n + 1/2*pi/ n)^4 + e^a)*sgn(4*x^4*abs(c)^(4/n)*cos(-1/2*pi*sgn(c)/n + 1/2*pi/n)^3*e^(3 *a)*sin(-1/2*pi*sgn(c)/n + 1/2*pi/n) - 4*x^4*abs(c)^(4/n)*cos(-1/2*pi*sgn( c)/n + 1/2*pi/n)*e^(3*a)*sin(-1/2*pi*sgn(c)/n + 1/2*pi/n)^3) - 1/2*pi*sgn( 4*x^4*abs(c)^(4/n)*cos(-1/2*pi*sgn(c)/n + 1/2*pi/n)^3*e^(3*a)*sin(-1/2*pi* sgn(c)/n + 1/2*pi/n) - 4*x^4*abs(c)^(4/n)*cos(-1/2*pi*sgn(c)/n + 1/2*pi/n) *e^(3*a)*sin(-1/2*pi*sgn(c)/n + 1/2*pi/n)^3) - arctan(-4*x^4*abs(c)^(4/n)* e^(2*a)*tan(-1/2*pi*sgn(c)/n + 1/2*pi/n)^3/(x^4*abs(c)^(4/n)*e^(2*a)*tan(- 1/2*pi*sgn(c)/n + 1/2*pi/n)^4 - 6*x^4*abs(c)^(4/n)*e^(2*a)*tan(-1/2*pi*sgn (c)/n + 1/2*pi/n)^2 + x^4*abs(c)^(4/n)*e^(2*a) + tan(-1/2*pi*sgn(c)/n + 1/ 2*pi/n)^4 + 2*tan(-1/2*pi*sgn(c)/n + 1/2*pi/n)^2 + 1) + 4*x^4*abs(c)^(4/n) *e^(2*a)*tan(-1/2*pi*sgn(c)/n + 1/2*pi/n)/(x^4*abs(c)^(4/n)*e^(2*a)*tan(-1 /2*pi*sgn(c)/n + 1/2*pi/n)^4 - 6*x^4*abs(c)^(4/n)*e^(2*a)*tan(-1/2*pi*sgn( c)/n + 1/2*pi/n)^2 + x^4*abs(c)^(4/n)*e^(2*a) + tan(-1/2*pi*sgn(c)/n + 1/2 *pi/n)^4 + 2*tan(-1/2*pi*sgn(c)/n + 1/2*pi/n)^2 + 1)))*e^a - 2*(x^8*abs...
Timed out. \[ \int \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{5/2} \,d x \]