Integrand size = 19, antiderivative size = 62 \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} e^{-a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4572, 4578} \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} x e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{2} x e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
[In]
[Out]
Rule 4572
Rule 4578
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 \left (a+\sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (\frac {e^{a \sqrt {-\frac {1}{n^2}} n}}{x}+e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{n}}\right ) \, dx,x,c x^n\right )}{2 n} \\ & = \frac {1}{4} e^{-a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \\ \end{align*}
\[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
[In]
[Out]
\[\int \cos \left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.65 \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{4} \, {\left (x^{2} + 2 \, e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - \log \left (c\right )}{n}\right )} \]
[In]
[Out]
\[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {2}{n}} x^{2} \cos \left (a\right ) + 2 \, \cos \left (a\right ) \log \left (x\right )}{4 \, c^{\left (\frac {1}{n}\right )}} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.02 \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
[In]
[Out]
Time = 28.80 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34 \[ \int \cos \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{2\,n\,\sqrt {-\frac {1}{n^2}}+2{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,n\,\sqrt {-\frac {1}{n^2}}-2{}\mathrm {i}} \]
[In]
[Out]