Integrand size = 21, antiderivative size = 515 \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {2 a b^3 \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {a b^3 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))} \]
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Time = 1.03 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2976, 2729, 2727, 2743, 2833, 12, 2738, 214} \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {2 a b^3 \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{9/2} (a+b)^{9/2}}-\frac {a b^3 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{9/2} (a+b)^{9/2}}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}+\frac {3 a^2 b^4 \sin (c+d x)}{2 d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}-\frac {a^2 b^3 \sin (c+d x)}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}-\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{9/2} (a+b)^{9/2}}-\frac {\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))}-\frac {(a-2 b) \sin (c+d x)}{4 d (a+b)^4 (1-\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 d (a-b)^4 (\cos (c+d x)+1)}+\frac {\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)}-\frac {\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)^2} \]
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Rule 12
Rule 214
Rule 2727
Rule 2729
Rule 2738
Rule 2743
Rule 2833
Rule 2976
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc (c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \int \left (\frac {1}{4 (a-b)^3 (-1-\cos (c+d x))^2}+\frac {-a-2 b}{4 (a-b)^4 (-1-\cos (c+d x))}+\frac {1}{4 (a+b)^3 (1-\cos (c+d x))^2}+\frac {a-2 b}{4 (a+b)^4 (1-\cos (c+d x))}+\frac {a b^3}{\left (a^2-b^2\right )^2 (-b-a \cos (c+d x))^3}+\frac {a b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (-b-a \cos (c+d x))^2}+\frac {a b \left (3 a^4+8 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4 (-b-a \cos (c+d x))}\right ) \, dx \\ & = \frac {\int \frac {1}{(-1-\cos (c+d x))^2} \, dx}{4 (a-b)^3}+\frac {(a-2 b) \int \frac {1}{1-\cos (c+d x)} \, dx}{4 (a+b)^4}+\frac {\int \frac {1}{(1-\cos (c+d x))^2} \, dx}{4 (a+b)^3}-\frac {(a+2 b) \int \frac {1}{-1-\cos (c+d x)} \, dx}{4 (a-b)^4}+\frac {\left (a b^3\right ) \int \frac {1}{(-b-a \cos (c+d x))^3} \, dx}{\left (a^2-b^2\right )^2}+\frac {\left (a b^2 \left (3 a^2+b^2\right )\right ) \int \frac {1}{(-b-a \cos (c+d x))^2} \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (a b \left (3 a^4+8 a^2 b^2+b^4\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^4} \\ & = -\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}-\frac {\int \frac {1}{-1-\cos (c+d x)} \, dx}{12 (a-b)^3}+\frac {\int \frac {1}{1-\cos (c+d x)} \, dx}{12 (a+b)^3}+\frac {\left (a b^3\right ) \int \frac {2 b-a \cos (c+d x)}{(-b-a \cos (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )^3}+\frac {\left (a b^2 \left (3 a^2+b^2\right )\right ) \int \frac {b}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^4}+\frac {\left (2 a b \left (3 a^4+8 a^2 b^2+b^4\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 )^4 d} \\ & = -\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (a b^3\right ) \int \frac {a^2+2 b^2}{-b-a \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^4}+\frac {\left (a b^3 \left (3 a^2+b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^4} \\ & = -\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (a b^3 \left (a^2+2 b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^4}+\frac {\left (2 a 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 )}{\left (a^2-b^2\right )^4 d} \\ & = -\frac {2 a 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-b)^{9/2} (a+b)^{9/2} d}-\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (a 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 )}{\left (a^2-b^2\right )^4 d} \\ & = -\frac {2 a 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-b)^{9/2} (a+b)^{9/2} d}-\frac {a 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-b)^{9/2} (a+b)^{9/2} d}-\frac {2 a b \left (3 a^4+8 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{9/2} (a+b)^{9/2} d}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))^2}-\frac {(a-2 b) \sin (c+d x)}{4 (a+b)^4 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^3 d (1+\cos (c+d x))}+\frac {(a+2 b) \sin (c+d x)}{4 (a-b)^4 d (1+\cos (c+d x))}-\frac {a^2 b^3 \sin (c+d x)}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^4 \sin (c+d x)}{2 \left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {a^2 b^2 \left (3 a^2+b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.75 \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {96 a b \left (6 a^4+23 a^2 b^2+6 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2}{\sqrt {a^2-b^2}}+\left (36 a^6 b+154 a^4 b^3+424 a^2 b^5+16 b^7-2 a \left (16 a^6-94 a^4 b^2-35 a^2 b^4+8 b^6\right ) \cos (c+d x)+8 \left (2 a^6 b-45 a^4 b^3-56 a^2 b^5-6 b^7\right ) \cos (2 (c+d x))-4 a^7 \cos (3 (c+d x))-154 a^5 b^2 \cos (3 (c+d x))-205 a^3 b^4 \cos (3 (c+d x))+48 a b^6 \cos (3 (c+d x))-20 a^6 b \cos (4 (c+d x))+110 a^4 b^3 \cos (4 (c+d x))+120 a^2 b^5 \cos (4 (c+d x))+4 a^7 \cos (5 (c+d x))+62 a^5 b^2 \cos (5 (c+d x))+39 a^3 b^4 \cos (5 (c+d x))\right ) \csc ^3(c+d x)\right ) \sec ^3(c+d x)}{96 \left (a^2-b^2\right )^4 d (a+b \sec (c+d x))^3} \]
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Time = 1.26 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3 a -3 b}{8 \left (a +b \right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 a b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}+3 a \,b^{4}-3 a^{4} b -\frac {5}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}+3 a \,b^{4}+3 a^{4} b +\frac {5}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}+23 a^{2} b^{2}+6 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4}}}{d}\) | \(328\) |
default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{3}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (a -b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3 a -3 b}{8 \left (a +b \right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 a b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}+3 a \,b^{4}-3 a^{4} b -\frac {5}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}+3 a \,b^{4}+3 a^{4} b +\frac {5}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}+23 a^{2} b^{2}+6 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4}}}{d}\) | \(328\) |
risch | \(\text {Expression too large to display}\) | \(1137\) |
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Time = 0.39 (sec) , antiderivative size = 1550, normalized size of antiderivative = 3.01 \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.48 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.38 \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {24 \, {\left (6 \, a^{5} b + 23 \, a^{3} b^{3} + 6 \, a b^{5}\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^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 20 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} b^{4} \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} + b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9} - 9 \, a^{8} b + 36 \, a^{7} b^{2} - 84 \, a^{6} b^{3} + 126 \, a^{5} b^{4} - 126 \, a^{4} b^{5} + 84 \, a^{3} b^{6} - 36 \, a^{2} b^{7} + 9 \, a b^{8} - b^{9}} - \frac {24 \, {\left (6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\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 {9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 14.27 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d\,{\left (a-b\right )}^3}+\frac {\frac {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}{3\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^7-21\,a^6\,b+111\,a^5\,b^2-145\,a^4\,b^3+145\,a^3\,b^4-111\,a^2\,b^5+21\,a\,b^6-3\,b^7\right )}{{\left (a+b\right )}^4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (17\,a^6-102\,a^5\,b+399\,a^4\,b^2-364\,a^3\,b^3+399\,a^2\,b^4-102\,a\,b^5+17\,b^6\right )}{3\,{\left (a+b\right )}^3}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a-b\right )\,\left (a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4\right )}{3\,{\left (a+b\right )}^2}}{d\,\left (\left (-8\,a^6+48\,a^5\,b-120\,a^4\,b^2+160\,a^3\,b^3-120\,a^2\,b^4+48\,a\,b^5-8\,b^6\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (16\,a^6-64\,a^5\,b+80\,a^4\,b^2-80\,a^2\,b^4+64\,a\,b^5-16\,b^6\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-8\,a^6+16\,a^5\,b+8\,a^4\,b^2-32\,a^3\,b^3+8\,a^2\,b^4+16\,a\,b^5-8\,b^6\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a+b\right )}{8\,d\,{\left (a-b\right )}^4}+\frac {a\,b\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^8-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^6\,b^2+6{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^4-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^6+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^8}{{\left (a+b\right )}^{9/2}\,{\left (a-b\right )}^{7/2}}\right )\,\left (6\,a^4+23\,a^2\,b^2+6\,b^4\right )\,1{}\mathrm {i}}{d\,{\left (a+b\right )}^{9/2}\,{\left (a-b\right )}^{9/2}} \]
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