Integrand size = 19, antiderivative size = 93 \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {6 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3854, 3856, 2719} \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]
[In]
[Out]
Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sec ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {\sec (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n} \\ & = \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\left (3 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \sqrt {\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n} \\ & = \frac {6 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \left (12 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sin \left (a+b \log \left (c x^n\right )\right )+\sin \left (3 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{10 b n} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(119)=238\).
Time = 3.02 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.01
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{6}+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{5 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(280\) |
default | \(-\frac {2 \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{6}+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right )}{5 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\) | \(280\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{\frac {3}{2}} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) - 3 i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{5 \, b n} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]
[In]
[Out]