Integrand size = 19, antiderivative size = 130 \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^{1+m} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2 i+2 i m+3 b n}{4 b n},-\frac {2 i+2 i m-b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4582, 4580, 371} \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-3\right ),-\frac {2 i m-b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(-3 i b n+2 m+2) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rule 371
Rule 4580
Rule 4582
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \cos ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^{1+m} \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1+m}{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+m}{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^{1+m} \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 (1+m)}{b n}\right ),-\frac {2 i+2 i m-b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \\ \end{align*}
Time = 1.88 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.57 \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m} \left (6 b^2 n^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(2+2 m+i b n) \left (4 (1+m) \cos ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{(2+2 m+i b n) (2+2 m-3 i b n) (2+2 m+3 i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \]
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\[\int x^{m} {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
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\[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^m\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \]
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