Integrand size = 23, antiderivative size = 23 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2},x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \\ \end{align*}
Not integrable
Time = 47.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]
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Not integrable
Time = 1.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int \frac {\tan ^{m}\left (d x +c \right )}{{\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{2}}d x\]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 107.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 3.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 15.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^2} \,d x \]
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