Integrand size = 19, antiderivative size = 130 \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-7 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.11 (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} \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{m+1} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b n}\right ),-\frac {2 i m-7 b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(3 i b n+2 m+2) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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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 \frac {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}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {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 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ & = \frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \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-7 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) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(487\) vs. \(2(130)=260\).
Time = 3.98 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.75 \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {x^{1+m-i b n} \left (\left (4+8 m+4 m^2+b^2 n^2\right ) x^{2 i b n} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3 i b n}{2}\right )}{2 b n},-\frac {2 i+2 i m-7 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i-2 i m+3 b n) \left ((-2 i-2 i m+b n) \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )-2 x^{i b n} \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} (b n \cos (b n \log (x))-2 (1+m) \sin (b n \log (x)))\right )\right )}{b n (-2 i-2 i m+3 b n) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (-2 (1+m) \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+b n \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \]
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\[\int \frac {x^{m}}{{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x^{m}}{\cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x^{m}}{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x^m}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x^m}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \,d x \]
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