Integrand size = 19, antiderivative size = 67 \[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n} \] Output:
-6/5*I*EllipticE(I*sinh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/b/n+2/5*cosh(a+b*l n(c*x^n))^(3/2)*sinh(a+b*ln(c*x^n))/b/n
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{5 b n} \] Input:
Integrate[Cosh[a + b*Log[c*x^n]]^(5/2)/x,x]
Output:
((-6*I)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2] + Sqrt[Cosh[a + b*Log[c*x^n ]]]*Sinh[2*(a + b*Log[c*x^n])])/(5*b*n)
Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3039, 3042, 3115, 3042, 3119}
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.
| \(\displaystyle \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{5/2}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{5} \int \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b}+\frac {3}{5} \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b}-\frac {6 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b}}{n}\) |
Input:
Int[Cosh[a + b*Log[c*x^n]]^(5/2)/x,x]
Output:
((((-6*I)/5)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2])/b + (2*Cosh[a + b*Log [c*x^n]]^(3/2)*Sinh[a + b*Log[c*x^n]])/(5*b))/n
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(63)=126\).
Time = 7.94 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.82
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (8 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{7}-16 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{5}+10 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{3}-3 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{5 n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(256\) |
| default | \(\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (8 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{7}-16 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{5}+10 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{3}-3 \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{5 n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(256\) |
Input:
int(cosh(a+b*ln(c*x^n))^(5/2)/x,x,method=_RETURNVERBOSE)
Output:
2/5/n*((2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^ (1/2)*(8*cosh(1/2*a+1/2*b*ln(c*x^n))^7-16*cosh(1/2*a+1/2*b*ln(c*x^n))^5+10 *cosh(1/2*a+1/2*b*ln(c*x^n))^3-3*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(- 2*cosh(1/2*a+1/2*b*ln(c*x^n))^2+1)^(1/2)*EllipticE(cosh(1/2*a+1/2*b*ln(c*x ^n)),2^(1/2))-2*cosh(1/2*a+1/2*b*ln(c*x^n)))/(2*sinh(1/2*a+1/2*b*ln(c*x^n) )^4+sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/sinh(1/2*a+1/2*b*ln(c*x^n))/(2*co sh(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)/b
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (61) = 122\).
Time = 0.08 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.96 \[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {12 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) - {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 6 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 12 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 6 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 1\right )} \sqrt {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{10 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )}} \] Input:
integrate(cosh(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")
Output:
-1/10*(12*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*sqrt(2)*cosh(b*n* log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n* log(x) + b*log(c) + a)^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))) - (c osh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh( b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 6*(cosh (b*n*log(x) + b*log(c) + a)^2 - 2)*sinh(b*n*log(x) + b*log(c) + a)^2 - 12* cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 - 6*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) - 1)*s qrt(cosh(b*n*log(x) + b*log(c) + a)))/(b*n*cosh(b*n*log(x) + b*log(c) + a) ^2 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a)^2)
Timed out. \[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \] Input:
integrate(cosh(a+b*ln(c*x**n))**(5/2)/x,x)
Output:
Timed out
\[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \] Input:
integrate(cosh(a+b*log(c*x^n))^(5/2)/x,x, algorithm="maxima")
Output:
integrate(cosh(b*log(c*x^n) + a)^(5/2)/x, x)
\[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\cosh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \] Input:
integrate(cosh(a+b*log(c*x^n))^(5/2)/x,x, algorithm="giac")
Output:
integrate(cosh(b*log(c*x^n) + a)^(5/2)/x, x)
Timed out. \[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}}{x} \,d x \] Input:
int(cosh(a + b*log(c*x^n))^(5/2)/x,x)
Output:
int(cosh(a + b*log(c*x^n))^(5/2)/x, x)
\[ \int \frac {\cosh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sqrt {\cosh \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}\, {\cosh \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}}{x}d x \] Input:
int(cosh(a+b*log(c*x^n))^(5/2)/x,x)
Output:
int((sqrt(cosh(log(x**n*c)*b + a))*cosh(log(x**n*c)*b + a)**2)/x,x)