Integrand size = 19, antiderivative size = 97 \[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \] Output:
-2/3*I*cosh(a+b*ln(c*x^n))^(1/2)*InverseJacobiAM(1/2*I*(a+b*ln(c*x^n)),2^( 1/2))*sech(a+b*ln(c*x^n))^(1/2)/b/n+2/3*sinh(a+b*ln(c*x^n))/b/n/sech(a+b*l n(c*x^n))^(1/2)
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \left (-2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b n} \] Input:
Integrate[1/(x*Sech[a + b*Log[c*x^n]]^(3/2)),x]
Output:
(Sqrt[Sech[a + b*Log[c*x^n]]]*((-2*I)*Sqrt[Cosh[a + b*Log[c*x^n]]]*Ellipti cF[(I/2)*(a + b*Log[c*x^n]), 2] + Sinh[2*(a + b*Log[c*x^n])]))/(3*b*n)
Time = 0.39 (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, 3120}
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.
| \(\displaystyle \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \frac {1}{\text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{3/2}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\frac {1}{3} \int \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}+\frac {1}{3} \int \sqrt {\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{3} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )+\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}+\frac {1}{3} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 \sinh \left (a+b \log \left (c x^n\right )\right )}{3 b \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}-\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b}}{n}\) |
Input:
Int[1/(x*Sech[a + b*Log[c*x^n]]^(3/2)),x]
Output:
((((-2*I)/3)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(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]])/(3*b*S qrt[Sech[a + b*Log[c*x^n]]]))/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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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. \(236\) vs. \(2(85)=170\).
Time = 1.56 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.44
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (4 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{5}-6 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{3}+\sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{3 n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(237\) |
| default | \(\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (4 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{5}-6 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{3}+\sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{3 n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(237\) |
Input:
int(1/x/sech(a+b*ln(c*x^n))^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3/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)*(4*cosh(1/2*a+1/2*b*ln(c*x^n))^5-6*cosh(1/2*a+1/2*b*ln(c*x^n))^3+(-s inh(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)*EllipticF(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*cosh(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. 370 vs. \(2 (84) = 168\).
Time = 0.12 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.81 \[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {2} {\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 )^{3} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 6 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 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} - 1\right )} \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}} + 4 \, {\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 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 )}{6 \, {\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(1/x/sech(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")
Output:
1/6*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*lo g(c) + a)^3*sinh(b*n*log(x) + b*log(c) + a) + 6*cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log(c) + a)^2 + 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 - 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*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)) + 4*(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)*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)^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)
\[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \operatorname {sech}^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \] Input:
integrate(1/x/sech(a+b*ln(c*x**n))**(3/2),x)
Output:
Integral(1/(x*sech(a + b*log(c*x**n))**(3/2)), x)
\[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x/sech(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")
Output:
integrate(1/(x*sech(b*log(c*x^n) + a)^(3/2)), x)
Timed out. \[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x/sech(a+b*log(c*x^n))^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \] Input:
int(1/(x*(1/cosh(a + b*log(c*x^n)))^(3/2)),x)
Output:
int(1/(x*(1/cosh(a + b*log(c*x^n)))^(3/2)), x)
\[ \int \frac {1}{x \text {sech}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{{\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2} x}d x \] Input:
int(1/x/sech(a+b*log(c*x^n))^(3/2),x)
Output:
int(sqrt(sech(log(x**n*c)*b + a))/(sech(log(x**n*c)*b + a)**2*x),x)