Integrand size = 21, antiderivative size = 109 \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} m \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f (1+m)} \]
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Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3883, 3913, 3912, 71} \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {2^{m+\frac {1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac {1}{2}} (a \csc (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f (m+1)}-\frac {\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)} \]
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Rule 71
Rule 3883
Rule 3912
Rule 3913
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac {m \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx}{1+m} \\ & = -\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac {\left (m (1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int \csc (e+f x) (1+\csc (e+f x))^m \, dx}{1+m} \\ & = -\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}+\frac {\left (m \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m\right ) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x}} \, dx,x,\csc (e+f x)\right )}{f (1+m) \sqrt {1-\csc (e+f x)}} \\ & = -\frac {\cot (e+f x) (a+a \csc (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} m \cot (e+f x) (1+\csc (e+f x))^{-\frac {1}{2}-m} (a+a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x))\right )}{f (1+m)} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.16 \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=-\frac {(a (1+\csc (e+f x)))^m \left ((-1+m) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-1-m,-2 m,-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right )+(1+m) \operatorname {Hypergeometric2F1}\left (1-m,-2 m,2-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right ) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )^{-2 m}}{2 f (-1+m) (1+m)} \]
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\[\int \csc \left (f x +e \right )^{2} \left (a +a \csc \left (f x +e \right )\right )^{m}d x\]
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\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
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\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int \left (a \left (\csc {\left (e + f x \right )} + 1\right )\right )^{m} \csc ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
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\[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int { {\left (a \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx=\int \frac {{\left (a+\frac {a}{\sin \left (e+f\,x\right )}\right )}^m}{{\sin \left (e+f\,x\right )}^2} \,d x \]
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