Integrand size = 19, antiderivative size = 97 \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {6 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \] Output:
-6/5*I*cosh(a+b*ln(c*x^n))^(1/2)*EllipticE(I*sinh(1/2*a+1/2*b*ln(c*x^n)),2 ^(1/2))*sech(a+b*ln(c*x^n))^(1/2)/b/n+2/5*sinh(a+b*ln(c*x^n))/b/n/sech(a+b *ln(c*x^n))^(3/2)
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \left (-12 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sinh \left (a+b \log \left (c x^n\right )\right )+\sinh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{10 b n} \] Input:
Integrate[1/(x*Sech[a + b*Log[c*x^n]]^(5/2)),x]
Output:
(Sqrt[Sech[a + b*Log[c*x^n]]]*((-12*I)*Sqrt[Cosh[a + b*Log[c*x^n]]]*Ellipt icE[(I/2)*(a + b*Log[c*x^n]), 2] + Sinh[a + b*Log[c*x^n]] + Sinh[3*(a + b* Log[c*x^n])]))/(10*b*n)
Time = 0.38 (sec) , antiderivative size = 95, 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, 4256, 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.
| \(\displaystyle \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \frac {1}{\text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{5/2}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\frac {3}{5} \int \frac {1}{\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {3}{5} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {3}{5} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {6 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b}}{n}\) |
Input:
Int[1/(x*Sech[a + b*Log[c*x^n]]^(5/2)),x]
Output:
((((-6*I)/5)*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*Sinh[a + b*Log[c*x^n]])/(5*b*S ech[a + b*Log[c*x^n]]^(3/2)))/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[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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(89)=178\).
Time = 3.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.64
| 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(1/x/sech(a+b*ln(c*x^n))^(5/2),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. 602 vs. \(2 (87) = 174\).
Time = 0.25 (sec) , antiderivative size = 602, normalized size of antiderivative = 6.21 \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x/sech(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")
Output:
1/20*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^6 + 6*cosh(b*n*log(x) + b*l og(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^5 + sinh(b*n*log(x) + b*log(c) + a)^6 + (15*cosh(b*n*log(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*l og(c) + a)^4 - 11*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*(5*cosh(b*n*log(x) + b*log(c) + a)^3 - 11*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + (15*cosh(b*n*log(x) + b*log(c) + a)^4 - 66*cosh(b*n*log (x) + b*log(c) + a)^2 - 13)*sinh(b*n*log(x) + b*log(c) + a)^2 - 13*cosh(b* n*log(x) + b*log(c) + a)^2 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^5 - 22*c osh(b*n*log(x) + b*log(c) + a)^3 - 13*cosh(b*n*log(x) + b*log(c) + a))*sin h(b*n*log(x) + b*log(c) + a) - 1)*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*co sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*l og(x) + b*log(c) + a)^2 + 1)) - 24*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a )^3 + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log( c) + a) + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*lo g(c) + a)^2 + sqrt(2)*sinh(b*n*log(x) + b*log(c) + a)^3)*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*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n *cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log(c) + a) + 3*b*n *cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*...
Timed out. \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x/sech(a+b*ln(c*x**n))**(5/2),x)
Output:
Timed out
\[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/x/sech(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")
Output:
integrate(1/(x*sech(b*log(c*x^n) + a)^(5/2)), x)
Timed out. \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x/sech(a+b*log(c*x^n))^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \] Input:
int(1/(x*(1/cosh(a + b*log(c*x^n)))^(5/2)),x)
Output:
int(1/(x*(1/cosh(a + b*log(c*x^n)))^(5/2)), x)
\[ \int \frac {1}{x \text {sech}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{{\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} x}d x \] Input:
int(1/x/sech(a+b*log(c*x^n))^(5/2),x)
Output:
int(sqrt(sech(log(x**n*c)*b + a))/(sech(log(x**n*c)*b + a)**3*x),x)