Integrand size = 21, antiderivative size = 376 \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {2 b^3 \left (3 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{7/2} (a+b)^{7/2} d}-\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b^3 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))} \]
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Time = 1.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2976, 2727, 2743, 12, 2738, 214, 2833} \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {2 b^3 \left (3 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d (a-b)^{7/2} (a+b)^{7/2}}-\frac {b^3 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d (a-b)^{7/2} (a+b)^{7/2}}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {3 b^4 \sin (c+d x)}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {b^3 \sin (c+d x)}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x)}{2 d (a+b)^3 (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 d (a-b)^3 (\cos (c+d x)+1)} \]
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Rule 12
Rule 214
Rule 2727
Rule 2738
Rule 2743
Rule 2833
Rule 2976
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \cot ^2(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \int \left (-\frac {1}{2 (a-b)^3 (-1-\cos (c+d x))}+\frac {1}{2 (a+b)^3 (1-\cos (c+d x))}+\frac {3 a^2 b^2-b^4}{a \left (a^2-b^2\right )^2 (-b-a \cos (c+d x))^2}+\frac {a \left (3 a^2 b+b^3\right )}{\left (a^2-b^2\right )^3 (-b-a \cos (c+d x))}+\frac {b^3}{a \left (-a^2+b^2\right ) (b+a \cos (c+d x))^3}\right ) \, dx \\ & = -\frac {\int \frac {1}{-1-\cos (c+d x)} \, dx}{2 (a-b)^3}+\frac {\int \frac {1}{1-\cos (c+d x)} \, dx}{2 (a+b)^3}-\frac {b^3 \int \frac {1}{(b+a \cos (c+d x))^3} \, dx}{a \left (a^2-b^2\right )}+\frac {\left (b^2 \left (3 a^2-b^2\right )\right ) \int \frac {1}{(-b-a \cos (c+d x))^2} \, dx}{a \left (a^2-b^2\right )^2}+\frac {\left (a b \left (3 a^2+b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^3} \\ & = -\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {b^3 \int \frac {-2 b+a \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (3 a^2-b^2\right )\right ) \int \frac {b}{-b-a \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )^3}+\frac {\left (2 a b \left (3 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d} \\ & = -\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {b^3 \int \frac {a^2+2 b^2}{b+a \cos (c+d x)} \, dx}{2 a \left (a^2-b^2\right )^3}+\frac {\left (b^3 \left (3 a^2-b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )^3} \\ & = -\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\left (b^3 \left (a^2+2 b^2\right )\right ) \int \frac {1}{b+a \cos (c+d x)} \, dx}{2 a \left (a^2-b^2\right )^3}+\frac {\left (2 b^3 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d} \\ & = -\frac {2 b^3 \left (3 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{7/2} (a+b)^{7/2} d}-\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\left (b^3 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d} \\ & = -\frac {2 b^3 \left (3 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{7/2} (a+b)^{7/2} d}-\frac {2 a b \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {b^3 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{2 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{2 (a-b)^3 d (1+\cos (c+d x))}-\frac {b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {b^2 \left (3 a^2-b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.61 \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) \left (\frac {6 a b \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2}{\left (a^2-b^2\right )^{7/2}}-\frac {(b+a \cos (c+d x))^2 \cot \left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {b^3 \sin (c+d x)}{(a-b)^2 (a+b)^2}+\frac {b^2 \left (6 a^2+b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^3 (a+b)^3}+\frac {(b+a \cos (c+d x))^2 \tan \left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}\right )}{2 d (a+b \sec (c+d x))^3} \]
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Time = 0.91 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}-6 a^{2} b +6 a \,b^{2}-2 b^{3}}-\frac {1}{2 \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (\frac {\left (-3 a^{3} b +\frac {5}{2} a^{2} b^{2}-\frac {1}{2} a \,b^{3}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b +\frac {5}{2} a^{2} b^{2}+\frac {1}{2} a \,b^{3}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{2}+3 b^{2}\right ) a \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}}{d}\) | \(234\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}-6 a^{2} b +6 a \,b^{2}-2 b^{3}}-\frac {1}{2 \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (\frac {\left (-3 a^{3} b +\frac {5}{2} a^{2} b^{2}-\frac {1}{2} a \,b^{3}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b +\frac {5}{2} a^{2} b^{2}+\frac {1}{2} a \,b^{3}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{2}+3 b^{2}\right ) a \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}}{d}\) | \(234\) |
risch | \(-\frac {i \left (6 a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}+9 a^{3} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-2 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+24 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+21 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+2 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+4 a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+14 a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+12 a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-4 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-28 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-2 a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-39 a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-4 a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}-2 a^{6}-12 a^{4} b^{2}-a^{2} b^{4}\right )}{a \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (-a^{2}+b^{2}\right )^{3} d}-\frac {3 b \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {9 b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {3 b \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {9 b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}\) | \(674\) |
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Time = 0.33 (sec) , antiderivative size = 841, normalized size of antiderivative = 2.24 \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\left [\frac {22 \, a^{4} b^{3} - 14 \, a^{2} b^{5} - 8 \, b^{7} - 2 \, {\left (2 \, a^{7} + 10 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (2 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) \sin \left (d x + c\right ) + 2 \, {\left (2 \, a^{6} b - 17 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (16 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )}{4 \, {\left ({\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d\right )} \sin \left (d x + c\right )}, \frac {11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7} - {\left (2 \, a^{7} + 10 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (2 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{6} b - 17 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (16 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.41 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {6 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {2 \, {\left (6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}} - \frac {1}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
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Time = 15.35 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,{\left (a-b\right )}^3}-\frac {\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{a+b}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^5-5\,a^4\,b+22\,a^3\,b^2-20\,a^2\,b^3+7\,a\,b^4-5\,b^5\right )}{{\left (a+b\right )}^3}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^4-4\,a^3\,b+12\,a^2\,b^2-5\,a\,b^3+3\,b^4\right )}{{\left (a+b\right )}^2}}{d\,\left (\left (2\,a^5-10\,a^4\,b+20\,a^3\,b^2-20\,a^2\,b^3+10\,a\,b^4-2\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,a^5+12\,a^4\,b-8\,a^3\,b^2-8\,a^2\,b^3+12\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^5-2\,a^4\,b-4\,a^3\,b^2+4\,a^2\,b^3+2\,a\,b^4-2\,b^5\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^6-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^2+3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}{{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{5/2}}\right )\,\left (2\,a^2+3\,b^2\right )\,3{}\mathrm {i}}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
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