Integrand size = 19, antiderivative size = 59 \[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \]
-2*(cos(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cos(1/2*a+1/2*b*ln(c*x^n))*Ellipti cE(sin(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/b/n+2*sin(a+b*ln(c*x^n))/b/n/cos(a+ b*ln(c*x^n))^(1/2)
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (-E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}\right )}{b n} \]
(2*(-EllipticE[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Log[c*x^n]]/Sqrt[Cos[a + b*Log[c*x^n]]]))/(b*n)
Time = 0.27 (sec) , antiderivative size = 57, 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, 3116, 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.
\(\displaystyle \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sin \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )^{3/2}}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}-\int \sqrt {\sin \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{b \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}-\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b}}{n}\) |
((-2*EllipticE[(a + b*Log[c*x^n])/2, 2])/b + (2*Sin[a + b*Log[c*x^n]])/(b* Sqrt[Cos[a + b*Log[c*x^n]]]))/n
3.2.19.3.1 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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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. \(249\) vs. \(2(93)=186\).
Time = 2.46 (sec) , antiderivative size = 250, normalized size of antiderivative = 4.24
method | result | size |
derivativedivides | \(-\frac {2 \left (-2 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(250\) |
default | \(-\frac {2 \left (-2 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(250\) |
-2/n*(-2*cos(1/2*a+1/2*b*ln(c*x^n))*(-2*sin(1/2*a+1/2*b*ln(c*x^n))^4+sin(1 /2*a+1/2*b*ln(c*x^n))^2)^(1/2)*sin(1/2*a+1/2*b*ln(c*x^n))^2+(sin(1/2*a+1/2 *b*ln(c*x^n))^2)^(1/2)*(-1+2*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-2*sin(1 /2*a+1/2*b*ln(c*x^n))^4+sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*EllipticE(cos( 1/2*a+1/2*b*ln(c*x^n)),2^(1/2)))/(-2*sin(1/2*a+1/2*b*ln(c*x^n))^4+sin(1/2* a+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a+1/2*b*ln(c*x^n))/(2*cos(1/2*a+1/2*b* ln(c*x^n))^2-1)^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.54 \[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + 2 \, \sqrt {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
(-I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)*weierstrassZeta(-4, 0, weierstr assPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*log(x) + b*l og(c) + a))) + I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) - I*sin(b*n *log(x) + b*log(c) + a))) + 2*sqrt(cos(b*n*log(x) + b*log(c) + a))*sin(b*n *log(x) + b*log(c) + a))/(b*n*cos(b*n*log(x) + b*log(c) + a))
\[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
\[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
Time = 27.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,\sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\,\sqrt {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}} \]