Integrand size = 29, antiderivative size = 295 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b^5 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{5/2} d}+\frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \]
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Time = 0.27 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2976, 3855, 3852, 8, 3853, 2727, 2743, 12, 2739, 632, 210} \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \cot (c+d x)}{a^3 d}+\frac {2 b^5 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{5/2}}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 d \left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {b^6 \cos (c+d x)}{a^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)} \]
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Rule 8
Rule 12
Rule 210
Rule 632
Rule 2727
Rule 2739
Rule 2743
Rule 2976
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a^2+3 b^2\right ) \csc (c+d x)}{a^4}-\frac {2 b \csc ^2(c+d x)}{a^3}+\frac {\csc ^3(c+d x)}{a^2}-\frac {1}{2 (a+b)^2 (-1+\sin (c+d x))}-\frac {1}{2 (a-b)^2 (1+\sin (c+d x))}+\frac {b^5}{a^3 \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {b^5 \left (5 a^2-3 b^2\right )}{a^4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \csc ^3(c+d x) \, dx}{a^2}-\frac {\int \frac {1}{1+\sin (c+d x)} \, dx}{2 (a-b)^2}-\frac {(2 b) \int \csc ^2(c+d x) \, dx}{a^3}-\frac {\int \frac {1}{-1+\sin (c+d x)} \, dx}{2 (a+b)^2}+\frac {\left (b^5 \left (5 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )^2}+\frac {b^5 \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^3 \left (a^2-b^2\right )}+\frac {\left (a^2+3 b^2\right ) \int \csc (c+d x) \, dx}{a^4} \\ & = -\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{2 a^2}+\frac {b^5 \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )^2}+\frac {(2 b) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}+\frac {\left (2 b^5 \left (5 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^2 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {b^5 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^2}-\frac {\left (4 b^5 \left (5 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^2 d} \\ & = \frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\left (2 b^5\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right )^2 d} \\ & = \frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac {\left (4 b^5\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \left (a^2-b^2\right )^2 d} \\ & = \frac {2 b^5 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{5/2} d}+\frac {2 b^5 \left (5 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^6 \cos (c+d x)}{a^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.90 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.21 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {6 b^5 \left (2 a^2-b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^4 \left (a^2-b^2\right )^{5/2} d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right )}{a^3 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^2 d}-\frac {3 \left (a^2+2 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {3 \left (a^2+2 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^2 d}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b^6 \cos (c+d x)}{a^3 (a-b)^2 (a+b)^2 d (a+b \sin (c+d x))}-\frac {b \tan \left (\frac {1}{2} (c+d x)\right )}{a^3 d} \]
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Time = 1.54 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {4 b^{5} \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(261\) |
default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {4 b^{5} \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(261\) |
risch | \(\frac {8 a^{4} b^{2}-8 a^{2} b^{4}+6 b^{6}-16 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-8 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+10 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{5} b \,{\mathrm e}^{7 i \left (d x +c \right )}+5 i a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}-8 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+3 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-13 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-9 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+16 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-3 i a \,b^{5} {\mathrm e}^{7 i \left (d x +c \right )}+9 i a \,b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+11 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-8 i a^{3} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}-4 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) \left (a^{2}-b^{2}\right )^{2} d \,a^{3}}-\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}+\frac {3 i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{4}}+\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}-\frac {3 i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{4} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}\) | \(877\) |
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Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (275) = 550\).
Time = 1.36 (sec) , antiderivative size = 1844, normalized size of antiderivative = 6.25 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.39 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.43 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {48 \, {\left (2 \, a^{2} b^{5} - b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {16 \, {\left (2 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{7} - a^{5} b^{2} - a b^{6}\right )}}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}} + \frac {12 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 13.95 (sec) , antiderivative size = 2302, normalized size of antiderivative = 7.80 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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