Integrand size = 29, antiderivative size = 314 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 a x}{b^4}+\frac {3 \sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^4 d}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 \sqrt {a^2-b^2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))} \]
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Time = 0.65 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2976, 3855, 3852, 8, 2718, 2743, 2833, 12, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a^3 d}+\frac {3 \left (2 a^2+b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 b^4 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (2 a^2+b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 d \sqrt {a^2-b^2}}+\frac {3 a x}{b^4}+\frac {\cos (c+d x)}{b^3 d} \]
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Rule 8
Rule 12
Rule 210
Rule 632
Rule 2718
Rule 2739
Rule 2743
Rule 2833
Rule 2976
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 a}{b^4}-\frac {3 b \csc (c+d x)}{a^4}+\frac {\csc ^2(c+d x)}{a^3}-\frac {\sin (c+d x)}{b^3}-\frac {\left (a^2-b^2\right )^3}{a^2 b^4 (a+b \sin (c+d x))^3}+\frac {2 \left (2 a^6-3 a^4 b^2+b^6\right )}{a^3 b^4 (a+b \sin (c+d x))^2}-\frac {3 \left (2 a^6-a^4 b^2-b^6\right )}{a^4 b^4 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {3 a x}{b^4}+\frac {\int \csc ^2(c+d x) \, dx}{a^3}-\frac {\int \sin (c+d x) \, dx}{b^3}-\frac {(3 b) \int \csc (c+d x) \, dx}{a^4}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a^2 b^4}-\frac {\left (3 \left (2 a^6-a^4 b^2-b^6\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^4 b^4}+\frac {\left (2 \left (2 a^6-3 a^4 b^2+b^6\right )\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^3 b^4} \\ & = \frac {3 a x}{b^4}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right )^2 \int \frac {-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 a^2 b^4}-\frac {\left (2 \left (2 a^6-3 a^4 b^2+b^6\right )\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^3 b^4 \left (-a^2+b^2\right )}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}-\frac {\left (6 \left (2 a^6-a^4 b^2-b^6\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 b^4 d} \\ & = \frac {3 a x}{b^4}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}-\left (2 \left (\frac {1}{a^2}-\frac {2 a^2}{b^4}+\frac {1}{b^2}\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx-\frac {\left (a^2-b^2\right ) \int \frac {2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 a^2 b^4}+\frac {\left (12 \left (2 a^6-a^4 b^2-b^6\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 b^4 d} \\ & = \frac {3 a x}{b^4}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 \sqrt {a^2-b^2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^2 b^4}-\frac {\left (4 \left (\frac {1}{a^2}-\frac {2 a^2}{b^4}+\frac {1}{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 )}{d} \\ & = \frac {3 a x}{b^4}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 \sqrt {a^2-b^2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}+\frac {\left (8 \left (\frac {1}{a^2}-\frac {2 a^2}{b^4}+\frac {1}{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 )}{d}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^2+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^2 b^4 d} \\ & = \frac {3 a x}{b^4}-\frac {4 \left (\frac {1}{a^2}-\frac {2 a^2}{b^4}+\frac {1}{b^2}\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 \sqrt {a^2-b^2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^2-b^2\right ) \left (2 a^2+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^2 b^4 d} \\ & = \frac {3 a x}{b^4}-\frac {4 \left (\frac {1}{a^2}-\frac {2 a^2}{b^4}+\frac {1}{b^2}\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}-\frac {\sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^4 d}-\frac {6 \left (2 a^6-a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^4 \sqrt {a^2-b^2} d}+\frac {3 b \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2+b^2\right ) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 5.58 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {6 a^5 c}{b^4}+\frac {6 a^5 d x}{b^4}+\frac {6 \left (-2 a^6+a^4 b^2-a^2 b^4+2 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2}}-a \cot \left (\frac {1}{2} (c+d x)\right )+6 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a \cos (c+d x) \left (6 a^5+a^3 b^2-5 a b^4-b \left (-9 a^4+a^2 b^2+4 b^4\right ) \sin (c+d x)+2 a^3 b^2 \sin ^2(c+d x)\right )}{b^3 (a+b \sin (c+d x))^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]
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Time = 1.95 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {3 a \,b^{2} \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (4 a^{6}+9 a^{4} b^{2}-3 a^{2} b^{4}-10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (13 a^{4}+a^{2} b^{2}-14 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-2 a^{6} b -\frac {a^{4} b^{3}}{2}+\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (2 a^{6}-a^{4} b^{2}+a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} b^{4}}+\frac {\frac {2 b}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+6 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(326\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {3 a \,b^{2} \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (4 a^{6}+9 a^{4} b^{2}-3 a^{2} b^{4}-10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (13 a^{4}+a^{2} b^{2}-14 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-2 a^{6} b -\frac {a^{4} b^{3}}{2}+\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (2 a^{6}-a^{4} b^{2}+a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} b^{4}}+\frac {\frac {2 b}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+6 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(326\) |
risch | \(\frac {3 a x}{b^{4}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b^{3} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{3} d}-\frac {i \left (3 i a^{3} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-24 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-4 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-14 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}+i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+10 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+21 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}-6 i a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}-10 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+18 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{4} b^{2}-a^{2} b^{4}-6 b^{6}\right )}{\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 )^{2} b^{4} a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{4} d}-\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,b^{2} a^{2}}-\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{4}}+\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}+\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{2 d \,b^{2} a^{2}}+\frac {3 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{4}}\) | \(747\) |
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Time = 0.62 (sec) , antiderivative size = 1171, normalized size of antiderivative = 3.73 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {6 \, {\left (d x + c\right )} a}{b^{4}} - \frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {6 \, {\left (2 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 2 \, b^{6}\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 )}}{\sqrt {a^{2} - b^{2}} a^{4} b^{4}} + \frac {2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a b^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{4} b^{3}} + \frac {2 \, {\left (3 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{6} + a^{4} b^{2} - 5 \, a^{2} b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{4} b^{3}}}{2 \, d} \]
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Time = 13.75 (sec) , antiderivative size = 4223, normalized size of antiderivative = 13.45 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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