Integrand size = 33, antiderivative size = 627 \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {b \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \left (-a^2+b^2\right )^{5/4} f g^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f g^2 \sqrt {\cos (e+f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]
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Time = 1.00 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2977, 2645, 331, 335, 304, 209, 212, 2650, 2716, 2721, 2719, 2775, 2946, 2780, 2886, 2884, 211, 214} \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f g^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac {b \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f g^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f g \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f g \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}-\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f g^2 \sqrt {\cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}} \]
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 331
Rule 335
Rule 2645
Rule 2650
Rule 2716
Rule 2719
Rule 2721
Rule 2775
Rule 2780
Rule 2884
Rule 2886
Rule 2946
Rule 2977
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b \csc (e+f x)}{a^2 (g \cos (e+f x))^{3/2}}+\frac {\csc ^2(e+f x)}{a (g \cos (e+f x))^{3/2}}+\frac {b^2}{a^2 (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))}\right ) \, dx \\ & = \frac {\int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a}-\frac {b \int \frac {\csc (e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a^2}+\frac {b^2 \int \frac {1}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a^2} \\ & = -\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {3 \int \frac {1}{(g \cos (e+f x))^{3/2}} \, dx}{2 a}-\frac {\left (2 b^2\right ) \int \frac {\sqrt {g \cos (e+f x)} \left (\frac {a^2}{2}+\frac {b^2}{2}+\frac {1}{2} a b \sin (e+f x)\right )}{a+b \sin (e+f x)} \, dx}{a^2 \left (a^2-b^2\right ) g^2}+\frac {b \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{g^2}\right )} \, dx,x,g \cos (e+f x)\right )}{a^2 f g} \\ & = -\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{a^2 f g^3}-\frac {3 \int \sqrt {g \cos (e+f x)} \, dx}{2 a g^2}-\frac {b^2 \int \sqrt {g \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) g^2}-\frac {b^4 \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{a^2 \left (a^2-b^2\right ) g^2} \\ & = -\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{g^2}} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 f g^3}+\frac {b^3 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a \left (a^2-b^2\right ) g}-\frac {b^3 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a \left (a^2-b^2\right ) g}-\frac {b^5 \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{a^2 \left (a^2-b^2\right ) f g}-\frac {\left (3 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{2 a g^2 \sqrt {\cos (e+f x)}}-\frac {\left (b^2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)}} \\ & = -\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f g^2 \sqrt {\cos (e+f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b \text {Subst}\left (\int \frac {1}{g-x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 f g}-\frac {b \text {Subst}\left (\int \frac {1}{g+x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 f g}-\frac {\left (2 b^5\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f g}+\frac {\left (b^3 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}-\frac {\left (b^3 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}} \\ & = -\frac {b \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}-\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f g^2 \sqrt {\cos (e+f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f g}-\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f g} \\ & = -\frac {b \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}+\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \left (-a^2+b^2\right )^{5/4} f g^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f g^{3/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {2 b}{a^2 f g \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {3 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f g^2 \sqrt {\cos (e+f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a \left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}-\frac {b^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {3 \sin (e+f x)}{a f g \sqrt {g \cos (e+f x)}}-\frac {2 b^2 (b-a \sin (e+f x))}{a^2 \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 26.49 (sec) , antiderivative size = 1635, normalized size of antiderivative = 2.61 \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {\cos ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (6 a^3+2 a b^2\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right )}{\sqrt {1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac {\left (7 a^2 b-5 b^3\right ) \left (-1+\cos ^2(e+f x)\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \csc (e+f x) \left (6 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-6 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+12 \left (a^2-b^2\right ) \arctan \left (\sqrt {\cos (e+f x)}\right )+8 a b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+6 a^2 \log \left (1-\sqrt {\cos (e+f x)}\right )-6 b^2 \log \left (1-\sqrt {\cos (e+f x)}\right )-6 a^2 \log \left (1+\sqrt {\cos (e+f x)}\right )+6 b^2 \log \left (1+\sqrt {\cos (e+f x)}\right )-3 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+3 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )}{12 \left (a^3-a b^2\right ) \left (1-\cos ^2(e+f x)\right ) (b+a \csc (e+f x))}-\frac {\left (-3 a^2 b+b^3\right ) \left (-1+\cos ^2(e+f x)\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \cos (2 (e+f x)) \csc (e+f x) \left (-42 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+42 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+84 b^{3/2} \left (a^2-b^2\right ) \arctan \left (\sqrt {\cos (e+f x)}\right )-56 a b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+48 a b^{5/2} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {7}{2}}(e+f x)+42 b^{3/2} \left (a^2-b^2\right ) \log \left (1-\sqrt {\cos (e+f x)}\right )+42 b^{3/2} \left (-a^2+b^2\right ) \log \left (1+\sqrt {\cos (e+f x)}\right )+21 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )-21 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )}{84 b^{3/2} \left (a^3-a b^2\right ) \left (1-\cos ^2(e+f x)\right ) \left (-1+2 \cos ^2(e+f x)\right ) (b+a \csc (e+f x))}\right )}{4 a (a-b) (a+b) f (g \cos (e+f x))^{3/2}}+\frac {\cos ^2(e+f x) \left (-\frac {\cot (e+f x)}{a}+\frac {2 \sec (e+f x) (-b+a \sin (e+f x))}{a^2-b^2}\right )}{f (g \cos (e+f x))^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.20 (sec) , antiderivative size = 1792, normalized size of antiderivative = 2.86
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Timed out. \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (g \cos {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \sin {\left (e + f x \right )}\right )}\, dx \]
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Exception generated. \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
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