Integrand size = 15, antiderivative size = 110 \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {2 i+b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4572, 4580, 371} \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]
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Rule 371
Rule 4572
Rule 4580
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cos ^{\frac {5}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{\frac {5 i b}{2}-\frac {1}{n}} \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {5 i b}{2}+\frac {1}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \\ & = \frac {2 x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {2 i+b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(696\) vs. \(2(110)=220\).
Time = 8.49 (sec) , antiderivative size = 696, normalized size of antiderivative = 6.33 \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {30 b^3 e^{i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} n^3 x^{1-i b n} \sqrt {2+2 e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}} \left ((2 i+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )+(-2 i+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )\right )}{(2-5 i b n) (2 i+b n) (-2 i+3 b n) (-2 i+5 b n) \left (-2 i-b n+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} (-2 i+b n)\right ) \sqrt {e^{-i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{-i b n} \left (1+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )}}+\sqrt {\cos \left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} \left (-\frac {2 x \left (2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+15 b^2 n^2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n) \left (-2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}+\frac {x \sin (2 b n \log (x)) \left (5 b n \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )-2 \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n)}+\frac {x \cos (2 b n \log (x)) \left (2 \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+5 b n \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n)}\right ) \]
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\[\int {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {5}{2}}d x\]
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Exception generated. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2} \,d x \]
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