Integrand size = 19, antiderivative size = 93 \[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
2*sinh(a+b*ln(c*x^n))*sech(a+b*ln(c*x^n))^(1/2)/b/n+2*I*(cosh(1/2*a+1/2*b* ln(c*x^n))^2)^(1/2)/cosh(1/2*a+1/2*b*ln(c*x^n))*EllipticE(I*sinh(1/2*a+1/2 *b*ln(c*x^n)),2^(1/2))*cosh(a+b*ln(c*x^n))^(1/2)*sech(a+b*ln(c*x^n))^(1/2) /b/n
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \left (i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
(2*Sqrt[Sech[a + b*Log[c*x^n]]]*(I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[ (I/2)*(a + b*Log[c*x^n]), 2] + Sinh[a + b*Log[c*x^n]]))/(b*n)
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3039, 3042, 4255, 3042, 4258, 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.
| \(\displaystyle \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{3/2}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b}-\int \frac {1}{\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b}-\int \frac {1}{\sqrt {\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b}-\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b}-\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b}+\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b}}{n}\) |
(((2*I)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2 ]*Sqrt[Sech[a + b*Log[c*x^n]]])/b + (2*Sqrt[Sech[a + b*Log[c*x^n]]]*Sinh[a + b*Log[c*x^n]])/b)/n
3.2.97.3.1 Defintions of rubi rules used
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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Time = 1.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(\frac {4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+2 \sqrt {-2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\) | \(141\) |
| default | \(\frac {4 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+2 \sqrt {-2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\) | \(141\) |
2/n*(2*cosh(1/2*a+1/2*b*ln(c*x^n))*sinh(1/2*a+1/2*b*ln(c*x^n))^2+(-2*sinh( 1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*E llipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2)))/sinh(1/2*a+1/2*b*ln(c*x^n)) /(-1+2*cosh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.71 \[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1}} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} + \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )\right )}}{b n} \]
2*(sqrt(2)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log (c) + a))/(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c ) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 + 1))*(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)) + sqrt(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))))/(b*n)
\[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
\[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]