Integrand size = 29, antiderivative size = 220 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^6 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{5/2} d}+\frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}+\frac {b \left (-a^2+2 b^2\right ) \sec (c+d x)}{\left (a^2-b^2\right )^2 d}+\frac {b \sec ^3(c+d x) (-a+b \sin (c+d x))}{3 a \left (a^2-b^2\right ) d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right )^2 d}+\frac {\tan ^3(c+d x)}{3 a d} \]
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Time = 0.33 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2977, 2702, 308, 213, 2700, 276, 2775, 2945, 12, 2739, 632, 210} \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^6 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{5/2}}+\frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 d \left (a^2-b^2\right )}+\frac {b^2 \sec (c+d x) \left (a \left (2 a^2-5 b^2\right ) \sin (c+d x)+3 b^3\right )}{3 a^2 d \left (a^2-b^2\right )^2}-\frac {b \sec ^3(c+d x)}{3 a^2 d}-\frac {b \sec (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {2 \tan (c+d x)}{a d}-\frac {\cot (c+d x)}{a d} \]
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Rule 12
Rule 210
Rule 213
Rule 276
Rule 308
Rule 632
Rule 2700
Rule 2702
Rule 2739
Rule 2775
Rule 2945
Rule 2977
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b \csc (c+d x) \sec ^4(c+d x)}{a^2}+\frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a}+\frac {b^2 \sec ^4(c+d x)}{a^2 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \csc ^2(c+d x) \sec ^4(c+d x) \, dx}{a}-\frac {b \int \csc (c+d x) \sec ^4(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \int \frac {\sec ^2(c+d x) \left (-2 a^2+3 b^2-2 a b \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d}+\frac {b^2 \int \frac {3 b^4}{a+b \sin (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )^2}+\frac {\text {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b \sec ^3(c+d x)}{3 a^2 d}-\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d}+\frac {2 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {b^6 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^2}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b \sec ^3(c+d x)}{3 a^2 d}-\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d}+\frac {2 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {\left (2 b^6\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 {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b \sec ^3(c+d x)}{3 a^2 d}-\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d}+\frac {2 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\left (4 b^6\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^6 \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 {b \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d}-\frac {b \sec (c+d x)}{a^2 d}-\frac {b \sec ^3(c+d x)}{3 a^2 d}-\frac {b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b^2 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d}+\frac {2 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(450\) vs. \(2(220)=440\).
Time = 6.81 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.05 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^6 \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{5/2} d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{2 a d}+\frac {b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d}-\frac {b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d}+\frac {1}{12 (a+b) d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{6 (a+b) d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{6 (a-b) d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {1}{12 (a-b) d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {10 a \sin \left (\frac {1}{2} (c+d x)\right )-13 b \sin \left (\frac {1}{2} (c+d x)\right )}{6 (a-b)^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {10 a \sin \left (\frac {1}{2} (c+d x)\right )+13 b \sin \left (\frac {1}{2} (c+d x)\right )}{6 (a+b)^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{2 a d} \]
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Time = 0.94 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {4 a -5 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 b^{6} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {4 a +5 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(253\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {4 a -5 b}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 b^{6} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {4 a +5 b}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(253\) |
risch | \(-\frac {2 \left (9 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}-6 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}+16 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-6 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+7 a^{3} {\mathrm e}^{5 i \left (d x +c \right )} b -10 b^{3} a \,{\mathrm e}^{5 i \left (d x +c \right )}-28 i a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i b^{4}+3 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-7 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+10 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-14 i a^{2} b^{2}+9 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+8 i a^{4}-3 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a \right )}{3 d \left (a^{2}-b^{2}\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}-\frac {b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}\) | \(493\) |
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Time = 0.83 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.78 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {3 \, \sqrt {-a^{2} + b^{2}} b^{6} \cos \left (d x + c\right )^{3} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) \sin \left (d x + c\right ) - 2 \, a^{7} + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} - 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 2 \, {\left (8 \, a^{7} - 22 \, a^{5} b^{2} + 17 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{7} - 11 \, a^{5} b^{2} + 7 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )}, -\frac {6 \, \sqrt {a^{2} - b^{2}} b^{6} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{7} + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} - 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 2 \, {\left (8 \, a^{7} - 22 \, a^{5} b^{2} + 17 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{7} - 11 \, a^{5} b^{2} + 7 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + 3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )}\right ] \]
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Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.43 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.62 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {12 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{6}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {3 \, {\left (2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {4 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{2} b + 7 \, b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 17.55 (sec) , antiderivative size = 2317, normalized size of antiderivative = 10.53 \[ \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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