Optimal. Leaf size=91 \[ \frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]
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Rubi [A] time = 0.15, antiderivative size = 91, 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} \[ \frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]
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 ^8(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^7(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^8(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^9(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^8 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\csc ^9(c+d x)}{9 a d}-\frac {\operatorname {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}-\frac {\csc ^9(c+d x)}{9 a d}\\ \end {align*}
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Mathematica [B] time = 1.06, size = 200, normalized size = 2.20 \[ -\frac {\csc (c) (-85750 \sin (c+d x)-17150 \sin (2 (c+d x))+51450 \sin (3 (c+d x))+17150 \sin (4 (c+d x))-17150 \sin (5 (c+d x))-7350 \sin (6 (c+d x))+2450 \sin (7 (c+d x))+1225 \sin (8 (c+d x))-28672 \sin (c+2 d x)+86016 \sin (2 c+3 d x)+28672 \sin (3 c+4 d x)-28672 \sin (4 c+5 d x)-12288 \sin (5 c+6 d x)+4096 \sin (6 c+7 d x)+2048 \sin (7 c+8 d x)+645120 \sin (c)-143360 \sin (d x)) \csc ^7(c+d x) \sec (c+d x)}{5160960 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 177, normalized size = 1.95 \[ -\frac {16 \, \cos \left (d x + c\right )^{8} + 16 \, \cos \left (d x + c\right )^{7} - 56 \, \cos \left (d x + c\right )^{6} - 56 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )^{2} - 35 \, \cos \left (d x + c\right ) - 35}{315 \, {\left (a d \cos \left (d x + c\right )^{7} + a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{5} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{3} + 3 \, a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 132, normalized size = 1.45 \[ -\frac {\frac {3 \, {\left (1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 126 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {35 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 270 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 882 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1470 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{9}}}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 114, normalized size = 1.25 \[ \frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {14 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{256 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 176, normalized size = 1.93 \[ -\frac {\frac {\frac {1470 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {882 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} + \frac {3 \, {\left (\frac {126 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {490 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1470 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 201, normalized size = 2.21 \[ -\frac {45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+882\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+270\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{80640\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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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