Optimal. Leaf size=109 \[ \frac {\cot ^{11}(c+d x)}{11 a d}+\frac {4 \cot ^9(c+d x)}{9 a d}+\frac {6 \cot ^7(c+d x)}{7 a d}+\frac {4 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^{11}(c+d x)}{11 a d} \]
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Rubi [A] time = 0.16, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3872, 2839, 2606, 30, 2607, 270} \[ \frac {\cot ^{11}(c+d x)}{11 a d}+\frac {4 \cot ^9(c+d x)}{9 a d}+\frac {6 \cot ^7(c+d x)}{7 a d}+\frac {4 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^{11}(c+d x)}{11 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^{10}(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^9(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^{11}(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^{10} \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\csc ^{11}(c+d x)}{11 a d}-\frac {\operatorname {Subst}\left (\int \left (x^2+4 x^4+6 x^6+4 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {4 \cot ^5(c+d x)}{5 a d}+\frac {6 \cot ^7(c+d x)}{7 a d}+\frac {4 \cot ^9(c+d x)}{9 a d}+\frac {\cot ^{11}(c+d x)}{11 a d}-\frac {\csc ^{11}(c+d x)}{11 a d}\\ \end {align*}
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Mathematica [B] time = 1.58, size = 242, normalized size = 2.22 \[ \frac {\csc (c) (5000940 \sin (c+d x)+833490 \sin (2 (c+d x))-3333960 \sin (3 (c+d x))-952560 \sin (4 (c+d x))+1428840 \sin (5 (c+d x))+535815 \sin (6 (c+d x))-357210 \sin (7 (c+d x))-158760 \sin (8 (c+d x))+39690 \sin (9 (c+d x))+19845 \sin (10 (c+d x))+1376256 \sin (c+2 d x)-5505024 \sin (2 c+3 d x)-1572864 \sin (3 c+4 d x)+2359296 \sin (4 c+5 d x)+884736 \sin (5 c+6 d x)-589824 \sin (6 c+7 d x)-262144 \sin (7 c+8 d x)+65536 \sin (8 c+9 d x)+32768 \sin (9 c+10 d x)-45416448 \sin (c)+8257536 \sin (d x)) \csc ^9(c+d x) \sec (c+d x)}{454164480 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 219, normalized size = 2.01 \[ -\frac {128 \, \cos \left (d x + c\right )^{10} + 128 \, \cos \left (d x + c\right )^{9} - 576 \, \cos \left (d x + c\right )^{8} - 576 \, \cos \left (d x + c\right )^{7} + 1008 \, \cos \left (d x + c\right )^{6} + 1008 \, \cos \left (d x + c\right )^{5} - 840 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 315 \, \cos \left (d x + c\right )^{2} + 315 \, \cos \left (d x + c\right ) + 315}{3465 \, {\left (a d \cos \left (d x + c\right )^{9} + a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{7} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{5} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{3} - 4 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 161, normalized size = 1.48 \[ -\frac {\frac {11 \, {\left (13230 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1701 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}} + \frac {315 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3080 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13365 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33264 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48510 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{11}}}{3548160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 140, normalized size = 1.28 \[ \frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {8 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {27 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}-\frac {16}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {27}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {42}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{1024 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 216, normalized size = 1.98 \[ -\frac {\frac {\frac {48510 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {33264 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {13365 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3080 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a} + \frac {11 \, {\left (\frac {360 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1701 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {5040 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {13230 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 35\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a \sin \left (d x + c\right )^{9}}}{3548160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 139, normalized size = 1.28 \[ -\frac {63\,\cos \left (c+d\,x\right )+\frac {21\,\cos \left (2\,c+2\,d\,x\right )}{2}-42\,\cos \left (3\,c+3\,d\,x\right )-12\,\cos \left (4\,c+4\,d\,x\right )+18\,\cos \left (5\,c+5\,d\,x\right )+\frac {27\,\cos \left (6\,c+6\,d\,x\right )}{4}-\frac {9\,\cos \left (7\,c+7\,d\,x\right )}{2}-2\,\cos \left (8\,c+8\,d\,x\right )+\frac {\cos \left (9\,c+9\,d\,x\right )}{2}+\frac {\cos \left (10\,c+10\,d\,x\right )}{4}+\frac {693}{2}}{3548160\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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