Integrand size = 21, antiderivative size = 274 \[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=-\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}+\frac {\sqrt {2} a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m}}{b^2 f (2+m) \sqrt {1+\csc (e+f x)}}-\frac {\sqrt {2} \left (a^2+b^2 (1+m)\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m}}{b^2 f (2+m) \sqrt {1+\csc (e+f x)}} \]
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Time = 0.42 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3925, 4092, 3919, 144, 143} \[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=-\frac {\sqrt {2} \left (a^2+b^2 (m+1)\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\csc (e+f x)+1}}+\frac {\sqrt {2} a (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m-1,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\csc (e+f x)+1}}-\frac {\cot (e+f x) (a+b \csc (e+f x))^{m+1}}{b f (m+2)} \]
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Rule 143
Rule 144
Rule 3919
Rule 3925
Rule 4092
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}+\frac {\int \csc (e+f x) (b (1+m)-a \csc (e+f x)) (a+b \csc (e+f x))^m \, dx}{b (2+m)} \\ & = -\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}-\frac {a \int \csc (e+f x) (a+b \csc (e+f x))^{1+m} \, dx}{b^2 (2+m)}+\frac {\left (a^2+b^2 (1+m)\right ) \int \csc (e+f x) (a+b \csc (e+f x))^m \, dx}{b^2 (2+m)} \\ & = -\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}-\frac {(a \cot (e+f x)) \text {Subst}\left (\int \frac {(a+b x)^{1+m}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\csc (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\csc (e+f x)} \sqrt {1+\csc (e+f x)}}+\frac {\left (\left (a^2+b^2 (1+m)\right ) \cot (e+f x)\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\csc (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\csc (e+f x)} \sqrt {1+\csc (e+f x)}} \\ & = -\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}+\frac {\left (a (-a-b) \cot (e+f x) (a+b \csc (e+f x))^m \left (-\frac {a+b \csc (e+f x)}{-a-b}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{1+m}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\csc (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\csc (e+f x)} \sqrt {1+\csc (e+f x)}}+\frac {\left (\left (a^2+b^2 (1+m)\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (-\frac {a+b \csc (e+f x)}{-a-b}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\csc (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\csc (e+f x)} \sqrt {1+\csc (e+f x)}} \\ & = -\frac {\cot (e+f x) (a+b \csc (e+f x))^{1+m}}{b f (2+m)}+\frac {\sqrt {2} a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m}}{b^2 f (2+m) \sqrt {1+\csc (e+f x)}}-\frac {\sqrt {2} \left (a^2+b^2 (1+m)\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\csc (e+f x)),\frac {b (1-\csc (e+f x))}{a+b}\right ) \cot (e+f x) (a+b \csc (e+f x))^m \left (\frac {a+b \csc (e+f x)}{a+b}\right )^{-m}}{b^2 f (2+m) \sqrt {1+\csc (e+f x)}} \\ \end{align*}
\[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx \]
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\[\int \csc \left (f x +e \right )^{3} \left (a +b \csc \left (f x +e \right )\right )^{m}d x\]
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\[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{3} \,d x } \]
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\[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int \left (a + b \csc {\left (e + f x \right )}\right )^{m} \csc ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{3} \,d x } \]
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\[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx=\int \frac {{\left (a+\frac {b}{\sin \left (e+f\,x\right )}\right )}^m}{{\sin \left (e+f\,x\right )}^3} \,d x \]
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