Integrand size = 21, antiderivative size = 21 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\text {Int}\left (\sin ^m(c+d x) (a+b \tan (c+d x))^n,x\right ) \]
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Not integrable
Time = 3.15 (sec) , antiderivative size = 21, 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 = {} \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx \\ \end{align*}
Not integrable
Time = 6.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx \]
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Not integrable
Time = 1.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \left (\sin ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}d x\]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]
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Not integrable
Time = 54.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sin ^{m}{\left (c + d x \right )}\, dx \]
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Not integrable
Time = 2.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]
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Not integrable
Time = 0.72 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]
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Not integrable
Time = 8.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx=\int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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