Integrand size = 34, antiderivative size = 34 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\frac {b (b x)^{1-n} \sin ^{-1+n}(a x)}{a^2 \left (a c^2 x \cos (a x)-c^2 \sin (a x)\right )}+\frac {b^2 (1-n) \text {Int}\left ((b x)^{-n} \sin ^{-2+n}(a x),x\right )}{a^2 c^2} \]
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Not integrable
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {b (b x)^{1-n} \sin ^{-1+n}(a x)}{a^2 \left (a c^2 x \cos (a x)-c^2 \sin (a x)\right )}+\frac {\left (b^2 (1-n)\right ) \int (b x)^{-n} \sin ^{-2+n}(a x) \, dx}{a^2 c^2} \\ \end{align*}
Not integrable
Time = 8.96 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
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Not integrable
Time = 0.96 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
\[\int \frac {\left (x b \right )^{2-n} \sin \left (a x \right )^{n}}{\left (a c x \cos \left (a x \right )-c \sin \left (a x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int { \frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{{\left (a c x \cos \left (a x\right ) - c \sin \left (a x\right )\right )}^{2}} \,d x } \]
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Not integrable
Time = 82.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\frac {\int \frac {\left (b x\right )^{2 - n} \sin ^{n}{\left (a x \right )}}{a^{2} x^{2} \cos ^{2}{\left (a x \right )} - 2 a x \sin {\left (a x \right )} \cos {\left (a x \right )} + \sin ^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
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Not integrable
Time = 2.80 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int { \frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{{\left (a c x \cos \left (a x\right ) - c \sin \left (a x\right )\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int { \frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{{\left (a c x \cos \left (a x\right ) - c \sin \left (a x\right )\right )}^{2}} \,d x } \]
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Not integrable
Time = 28.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx=\int \frac {{\sin \left (a\,x\right )}^n\,{\left (b\,x\right )}^{2-n}}{{\left (c\,\sin \left (a\,x\right )-a\,c\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
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