Integrand size = 15, antiderivative size = 109 \[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 109, 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} \[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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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 {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1}{n}} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b}{2}+\frac {1}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \\ & = \frac {2 x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(109)=218\).
Time = 0.80 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.02 \[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {6 i \sqrt {2} b^2 \sqrt {1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}} n^2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}-\frac {i}{2 b n},\frac {5}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (-2 i+b n) (-2 i+3 b n) (2 i+3 b n)}+\frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (2 \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+9 b^2 n^2} \]
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\[\int {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \]
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