Integrand size = 29, antiderivative size = 329 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \]
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Time = 0.96 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}+\frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
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Rule 210
Rule 632
Rule 2739
Rule 2975
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 \left (3 a^4+14 a^2 b^2-20 b^4\right )-6 a b \left (3 a^2-b^2\right ) \sin (c+d x)+6 b^2 \left (2 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{12 a^2 b^2} \\ & = -\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^2(c+d x) \left (-4 \left (3 a^6+17 a^4 b^2-50 a^2 b^4+30 b^6\right )-4 a b \left (6 a^4-11 a^2 b^2+5 b^4\right ) \sin (c+d x)+16 b^2 \left (3 a^4-8 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{24 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-4 \left (a^2-b^2\right )^2 \left (3 a^4+35 a^2 b^2-60 b^4\right )-4 a b \left (3 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+60 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (60 b^3 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right )^2+60 a b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^5 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\left (5 b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{2 a^6}+\frac {\left (5 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^6} \\ & = -\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac {\left (5 \left (a^2-4 b^2\right ) \left (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^6 d} \\ & = -\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac {\left (10 \left (a^2-4 b^2\right ) \left (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^6 d} \\ & = \frac {5 \left (a^2-4 b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {5 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (3 a^4+35 a^2 b^2-60 b^4\right ) \cot (c+d x)}{6 a^5 b^2 d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac {a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac {5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac {5 \left (a^2-2 b^2\right ) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.52 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.49 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}+\frac {\left (7 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-18 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^5 d}+\frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}-\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {5 \left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-7 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+18 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^5 d}+\frac {-a^4 \cos (c+d x)+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{2 a^4 b d (a+b \sin (c+d x))^2}+\frac {a^4 \cos (c+d x)+7 a^2 b^2 \cos (c+d x)-8 b^4 \cos (c+d x)}{2 a^5 b d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^3 d} \]
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Time = 1.07 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-9 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (-\frac {a \left (a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {9 b \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \left (a^{4}+25 a^{2} b^{2}-26 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {9 a^{4} b}{2}-\frac {9 a^{2} b^{3}}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {5 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(359\) |
default | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-9 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (-\frac {a \left (a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {9 b \left (a^{4}+a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \left (a^{4}+25 a^{2} b^{2}-26 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {9 a^{4} b}{2}-\frac {9 a^{2} b^{3}}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {5 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(359\) |
risch | \(\frac {-60 i b^{6}+36 a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}+15 a^{3} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-150 b^{3} a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+300 b^{5} a \,{\mathrm e}^{7 i \left (d x +c \right )}-24 b \,a^{5} {\mathrm e}^{3 i \left (d x +c \right )}+125 b^{3} a^{3} {\mathrm e}^{i \left (d x +c \right )}-210 b^{5} a \,{\mathrm e}^{i \left (d x +c \right )}+6 b \,a^{5} {\mathrm e}^{i \left (d x +c \right )}+6 i a^{6} {\mathrm e}^{8 i \left (d x +c \right )}-18 i a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+18 i a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{5} b \,{\mathrm e}^{9 i \left (d x +c \right )}-24 a^{5} b \,{\mathrm e}^{7 i \left (d x +c \right )}+240 i b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-60 i b^{6} {\mathrm e}^{8 i \left (d x +c \right )}+240 i b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-360 i b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-210 i b^{2} a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-110 i b^{2} a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+20 i a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+45 i a^{4} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-45 i b^{4} a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+180 i b^{4} a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-190 i b^{4} a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+240 i b^{2} a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+420 b^{3} a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-720 b^{5} {\mathrm e}^{5 i \left (d x +c \right )} a -30 b^{5} a \,{\mathrm e}^{9 i \left (d x +c \right )}+3 i a^{4} b^{2}+35 i a^{2} b^{4}-410 b^{3} a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+660 b^{5} a \,{\mathrm e}^{3 i \left (d x +c \right )}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} b^{2} a^{5} d}+\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{6} d}-\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{6} d}+\frac {5 \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 \,a^{4}}-\frac {10 \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 ) b^{2}}{d \,a^{6}}-\frac {5 \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 \,a^{4}}+\frac {10 \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 ) b^{2}}{d \,a^{6}}\) | \(878\) |
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Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (310) = 620\).
Time = 0.52 (sec) , antiderivative size = 1553, normalized size of antiderivative = 4.72 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.39 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {60 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {120 \, {\left (a^{4} - 5 \, a^{2} b^{2} + 4 \, b^{4}\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^{6}} - \frac {24 \, {\left (a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 25 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a^{4} b + 9 \, a^{2} b^{3}\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^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {330 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 11.98 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.29 \[ \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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