Optimal. Leaf size=125 \[ -\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d} \]
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Rubi [A] time = 0.37, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2875, 2873, 2607, 270, 2606, 14} \[ -\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc ^6(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\int (-a+a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^8(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^9(c+d x)+a^2 \cot ^2(c+d x) \csc ^{10}(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^{10}(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^3(c+d x) \csc ^9(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^4 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \operatorname {Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 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^2 d}+\frac {2 \operatorname {Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{a^2 d}-\frac {9 \cot ^7(c+d x)}{7 a^2 d}-\frac {7 \cot ^9(c+d x)}{9 a^2 d}-\frac {2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac {2 \csc ^9(c+d x)}{9 a^2 d}+\frac {2 \csc ^{11}(c+d x)}{11 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.50, size = 233, normalized size = 1.86 \[ -\frac {\csc (c) (-218834 \sin (c+d x)-79576 \sin (2 (c+d x))+119364 \sin (3 (c+d x))+79576 \sin (4 (c+d x))-28420 \sin (5 (c+d x))-34104 \sin (6 (c+d x))-1421 \sin (7 (c+d x))+5684 \sin (8 (c+d x))+1421 \sin (9 (c+d x))+1419264 \sin (2 c+d x)+114688 \sin (c+2 d x)-172032 \sin (2 c+3 d x)-114688 \sin (3 c+4 d x)+40960 \sin (4 c+5 d x)+49152 \sin (5 c+6 d x)+2048 \sin (6 c+7 d x)-8192 \sin (7 c+8 d x)-2048 \sin (8 c+9 d x)+630784 \sin (c)-1103872 \sin (d x)) \csc ^7(c+d x) \sec ^2(c+d x)}{22708224 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.54, size = 204, normalized size = 1.63 \[ \frac {16 \, \cos \left (d x + c\right )^{9} + 32 \, \cos \left (d x + c\right )^{8} - 40 \, \cos \left (d x + c\right )^{7} - 112 \, \cos \left (d x + c\right )^{6} + 14 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{3} - 70 \, \cos \left (d x + c\right )^{2} + 56 \, \cos \left (d x + c\right ) + 28}{693 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} + 2 \, a^{2} d \cos \left (d x + c\right )^{7} - 2 \, a^{2} d \cos \left (d x + c\right )^{6} - 6 \, a^{2} d \cos \left (d x + c\right )^{5} + 6 \, a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 134, normalized size = 1.07 \[ -\frac {\frac {33 \, {\left (56 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} - \frac {63 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 385 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 792 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3234 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9702 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{22}}}{354816 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 112, normalized size = 0.90 \[ \frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{512 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 174, normalized size = 1.39 \[ -\frac {\frac {\frac {9702 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3234 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {792 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2}} + \frac {33 \, {\left (\frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {56 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a^{2} \sin \left (d x + c\right )^{7}}}{354816 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 201, normalized size = 1.61 \[ -\frac {99\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1848\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9702\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3234\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-385\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-63\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{354816\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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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