Integrand size = 23, antiderivative size = 23 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\text {Int}\left (\left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x),x\right ) \]
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Not integrable
Time = 0.05 (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 = {} \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx \\ \end{align*}
Not integrable
Time = 3.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx \]
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Not integrable
Time = 0.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[\int {\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p} \left (\tan ^{2}\left (d x +c \right )\right )d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\text {Timed out} \]
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Not integrable
Time = 4.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{2} \,d x } \]
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Not integrable
Time = 6.73 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{2} \,d x } \]
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Not integrable
Time = 14.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^2(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]
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