Integrand size = 23, antiderivative size = 83 \[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,-p,\frac {3}{2},\cos ^2(e+f x),\frac {b \cos ^2(e+f x)}{a+b}\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p}}{f} \]
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Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 441, 440} \[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},2,-p,\frac {3}{2},\cos ^2(e+f x),\frac {b \cos ^2(e+f x)}{a+b}\right )}{f} \]
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Rule 440
Rule 441
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\left (\left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {b x^2}{a+b}\right )^p}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,-p,\frac {3}{2},\cos ^2(e+f x),\frac {b \cos ^2(e+f x)}{a+b}\right ) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p}}{f} \\ \end{align*}
\[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx \]
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\[\int \left (\csc ^{3}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p}d x\]
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\[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3} \,d x } \]
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\[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \csc \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \csc ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p}{{\sin \left (e+f\,x\right )}^3} \,d x \]
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