Integrand size = 28, antiderivative size = 133 \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {e^{\frac {a (1+m)}{\sqrt {-\frac {(1+m)^2}{n^2}} n}} x^{1+m} \left (c x^n\right )^{\frac {1+m}{n}}}{4 \sqrt {-\frac {(1+m)^2}{n^2}} n}+\frac {e^{\frac {a \sqrt {-\frac {(1+m)^2}{n^2}} n}{1+m}} (1+m) x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \log (x)}{2 \sqrt {-\frac {(1+m)^2}{n^2}} n} \]
[Out]
Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4581, 4577} \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {(m+1) x^{m+1} \log (x) e^{\frac {a n \sqrt {-\frac {(m+1)^2}{n^2}}}{m+1}} \left (c x^n\right )^{-\frac {m+1}{n}}}{2 n \sqrt {-\frac {(m+1)^2}{n^2}}}-\frac {x^{m+1} e^{\frac {a (m+1)}{n \sqrt {-\frac {(m+1)^2}{n^2}}}} \left (c x^n\right )^{\frac {m+1}{n}}}{4 n \sqrt {-\frac {(m+1)^2}{n^2}}} \]
[In]
[Out]
Rule 4577
Rule 4581
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}} \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left ((1+m) x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \left (\frac {e^{\frac {a \sqrt {-\frac {(1+m)^2}{n^2}} n}{1+m}}}{x}-e^{\frac {a (1+m)}{\sqrt {-\frac {(1+m)^2}{n^2}} n}} x^{-1+\frac {2 (1+m)}{n}}\right ) \, dx,x,c x^n\right )}{2 \sqrt {-\frac {(1+m)^2}{n^2}} n^2} \\ & = -\frac {e^{\frac {a (1+m)}{\sqrt {-\frac {(1+m)^2}{n^2}} n}} x^{1+m} \left (c x^n\right )^{\frac {1+m}{n}}}{4 \sqrt {-\frac {(1+m)^2}{n^2}} n}+\frac {e^{\frac {a \sqrt {-\frac {(1+m)^2}{n^2}} n}{1+m}} (1+m) x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \log (x)}{2 \sqrt {-\frac {(1+m)^2}{n^2}} n} \\ \end{align*}
\[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
[In]
[Out]
\[\int x^{m} \sin \left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {\left (1+m \right )^{2}}{n^{2}}}\right )d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.47 \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {{\left (i \, x^{2} x^{2 \, m} - 2 \, {\left (i \, m + i\right )} e^{\left (\frac {2 \, {\left (i \, a n - {\left (m + 1\right )} \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - {\left (m + 1\right )} \log \left (c\right )}{n}\right )}}{4 \, {\left (m + 1\right )}} \]
[In]
[Out]
\[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^{m} \sin {\left (a + \sqrt {- \frac {m^{2}}{n^{2}} - \frac {2 m}{n^{2}} - \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.62 \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {2 \, m}{n} + \frac {2}{n}} x e^{\left (m \log \left (x\right ) + \frac {m \log \left (x^{n}\right )}{n} + \frac {\log \left (x^{n}\right )}{n}\right )} \sin \left (a\right ) + 2 \, {\left (m \sin \left (a\right ) + \sin \left (a\right )\right )} \log \left (x\right )}{4 \, {\left (c^{\frac {m}{n} + \frac {1}{n}} m + c^{\frac {m}{n} + \frac {1}{n}}\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.05 \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {-i \, m n^{2} x x^{m} e^{\left (i \, a - \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + i \, m n^{2} x x^{m} e^{\left (-i \, a + \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - i \, n^{2} x x^{m} e^{\left (i \, a - \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - i \, n x x^{m} {\left | m n + n \right |} e^{\left (i \, a - \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + i \, n^{2} x x^{m} e^{\left (-i \, a + \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - i \, n x x^{m} {\left | m n + n \right |} e^{\left (-i \, a + \frac {n {\left | m n + n \right |} \log \left (x\right ) + {\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )}}{2 \, {\left (m^{2} n^{2} + 2 \, m n^{2} - {\left (m n + n\right )}^{2} + n^{2}\right )}} \]
[In]
[Out]
Time = 28.79 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02 \[ \int x^m \sin \left (a+\sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x\,x^m\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{2\,m+2-n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,2{}\mathrm {i}}-\frac {x\,x^m\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,m+2+n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,2{}\mathrm {i}} \]
[In]
[Out]