Integrand size = 29, antiderivative size = 395 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {x}{b^3}-\frac {6 \sqrt {a^2-b^2} \left (a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))} \]
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Time = 0.56 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2976, 3855, 3852, 8, 3853, 2743, 2833, 12, 2739, 632, 210} \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}-\frac {6 \left (a^2+b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\left (2 a^2+b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {6 \left (a^6+a^2 b^4-2 b^6\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 d \sqrt {a^2-b^2}}-\frac {x}{b^3} \]
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Rule 8
Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2743
Rule 2833
Rule 2976
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{b^3}-\frac {3 \left (a^2-2 b^2\right ) \csc (c+d x)}{a^5}-\frac {3 b \csc ^2(c+d x)}{a^4}+\frac {\csc ^3(c+d x)}{a^3}+\frac {\left (a^2-b^2\right )^3}{a^3 b^3 (a+b \sin (c+d x))^3}-\frac {3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )}{a^4 b^3 (a+b \sin (c+d x))^2}+\frac {3 \left (a^6+a^2 b^4-2 b^6\right )}{a^5 b^3 (a+b \sin (c+d x))}\right ) \, dx \\ & = -\frac {x}{b^3}+\frac {\int \csc ^3(c+d x) \, dx}{a^3}-\frac {(3 b) \int \csc ^2(c+d x) \, dx}{a^4}-\frac {\left (3 \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a^3 b^3}-\frac {\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^4 b^3}+\frac {\left (3 \left (a^6+a^2 b^4-2 b^6\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5 b^3} \\ & = -\frac {x}{b^3}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{2 a^3}-\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^3 b^3}+\frac {\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^4 b^3 \left (-a^2+b^2\right )}+\frac {(3 b) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}+\frac {\left (6 \left (a^6+a^2 b^4-2 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^5 b^3 d} \\ & = -\frac {x}{b^3}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}-\left (3 \left (\frac {a}{b^3}-\frac {b}{a^3}\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^3 b^3}-\frac {\left (12 \left (a^6+a^2 b^4-2 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^5 b^3 d} \\ & = -\frac {x}{b^3}+\frac {6 \left (a^6+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 )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 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^3 b^3}-\frac {\left (6 \left (\frac {a}{b^3}-\frac {b}{a^3}\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 {x}{b^3}+\frac {6 \left (a^6+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 )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac {\left (12 \left (\frac {a}{b^3}-\frac {b}{a^3}\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^3 b^3 d} \\ & = -\frac {x}{b^3}-\frac {6 \left (\frac {a}{b^3}-\frac {b}{a^3}\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 (a^6+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 )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 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^3 b^3 d} \\ & = -\frac {x}{b^3}-\frac {6 \left (\frac {a}{b^3}-\frac {b}{a^3}\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^3 b^3 d}+\frac {6 \left (a^6+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 )}{a^5 b^3 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \left (a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {3 b \cot (c+d x)}{a^4 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {3 \left (a^4-b^4\right ) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.54 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {c+d x}{b^3 d}+\frac {\left (2 a^6-a^4 b^2+11 a^2 b^4-12 b^6\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b^3 \sqrt {a^2-b^2} d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\left (5 a^2-12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\left (-5 a^2+12 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}-\frac {3 \left (a^4 \cos (c+d x)+a^2 b^2 \cos (c+d x)-2 b^4 \cos (c+d x)\right )}{2 a^4 b^2 d (a+b \sin (c+d x))}-\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]
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Time = 1.05 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+4 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (2 a^{6}+9 a^{4} b^{2}+3 a^{2} b^{4}-14 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (7 a^{4}+13 a^{2} b^{2}-20 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} b \left (2 a^{4}+5 a^{2} b^{2}-7 b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{6}-a^{4} b^{2}+11 a^{2} b^{4}-12 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5} b^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\) | \(357\) |
default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+4 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b \left (2 a^{6}+9 a^{4} b^{2}+3 a^{2} b^{4}-14 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (7 a^{4}+13 a^{2} b^{2}-20 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{2} b \left (2 a^{4}+5 a^{2} b^{2}-7 b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{6}-a^{4} b^{2}+11 a^{2} b^{4}-12 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5} b^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\) | \(357\) |
risch | \(-\frac {x}{b^{3}}+\frac {i \left (-3 a^{4} b^{2}-3 a^{2} b^{4}+12 b^{6}-20 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-27 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+90 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+8 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}+11 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-42 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}-12 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-4 i a^{5} b \,{\mathrm e}^{7 i \left (d x +c \right )}-i a^{3} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+6 i b^{5} a \,{\mathrm e}^{7 i \left (d x +c \right )}+9 a^{4} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-13 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-12 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+36 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-21 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+39 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+9 i a^{3} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-54 i a \,b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+16 i a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-36 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-23 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \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^{3} a^{4} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{5} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{5} d}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3} a}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{2 d b \,a^{3}}-\frac {6 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3} a}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{2 d b \,a^{3}}+\frac {6 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}\) | \(879\) |
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Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (372) = 744\).
Time = 0.71 (sec) , antiderivative size = 1658, normalized size of antiderivative = 4.20 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.39 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {8 \, {\left (d x + c\right )}}{b^{3}} + \frac {4 \, {\left (5 \, a^{2} - 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {8 \, {\left (2 \, a^{6} - a^{4} b^{2} + 11 \, a^{2} b^{4} - 12 \, 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^{5} b^{3}} - \frac {10 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 53 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 56 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 32 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 76 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4} b^{2}}{{\left (a \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 )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a^{5} b^{2}}}{8 \, d} \]
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Time = 12.88 (sec) , antiderivative size = 4381, normalized size of antiderivative = 11.09 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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