Optimal. Leaf size=284 \[ \frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac {a^2}{80 d (a \sin (c+d x)+a)^{10}}-\frac {7}{512 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac {a}{48 d (a \sin (c+d x)+a)^9}-\frac {3}{128 d (a \sin (c+d x)+a)^8}-\frac {5}{224 a d (a \sin (c+d x)+a)^7} \]
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Rubi [A] time = 0.22, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ -\frac {a^2}{80 d (a \sin (c+d x)+a)^{10}}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {7}{512 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac {21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac {3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {a}{48 d (a \sin (c+d x)+a)^9}-\frac {3}{128 d (a \sin (c+d x)+a)^8}-\frac {5}{224 a d (a \sin (c+d x)+a)^7} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {1}{2048 a^{11} (a-x)^3}+\frac {11}{4096 a^{12} (a-x)^2}+\frac {1}{8 a^3 (a+x)^{11}}+\frac {3}{16 a^4 (a+x)^{10}}+\frac {3}{16 a^5 (a+x)^9}+\frac {5}{32 a^6 (a+x)^8}+\frac {15}{128 a^7 (a+x)^7}+\frac {21}{256 a^8 (a+x)^6}+\frac {7}{128 a^9 (a+x)^5}+\frac {9}{256 a^{10} (a+x)^4}+\frac {45}{2048 a^{11} (a+x)^3}+\frac {55}{4096 a^{12} (a+x)^2}+\frac {33}{2048 a^{12} \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac {a}{48 d (a+a \sin (c+d x))^9}-\frac {3}{128 d (a+a \sin (c+d x))^8}-\frac {5}{224 a d (a+a \sin (c+d x))^7}-\frac {5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac {21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac {3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac {7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {33 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2048 a^7 d}\\ &=\frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac {a}{48 d (a+a \sin (c+d x))^9}-\frac {3}{128 d (a+a \sin (c+d x))^8}-\frac {5}{224 a d (a+a \sin (c+d x))^7}-\frac {5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac {21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac {3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac {7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 2.61, size = 195, normalized size = 0.69 \[ \frac {\sec ^4(c+d x) \left (-3465 \sin ^{11}(c+d x)-27720 \sin ^{10}(c+d x)-91245 \sin ^9(c+d x)-147840 \sin ^8(c+d x)-82698 \sin ^7(c+d x)+114576 \sin ^6(c+d x)+255222 \sin ^5(c+d x)+190080 \sin ^4(c+d x)+21395 \sin ^3(c+d x)-72776 \sin ^2(c+d x)-66953 \sin (c+d x)+3465 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{20}-34816\right )}{215040 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 466, normalized size = 1.64 \[ \frac {55440 \, \cos \left (d x + c\right )^{10} - 572880 \, \cos \left (d x + c\right )^{8} + 1507968 \, \cos \left (d x + c\right )^{6} - 1260864 \, \cos \left (d x + c\right )^{4} + 157696 \, \cos \left (d x + c\right )^{2} + 3465 \, {\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \, {\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \, {\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \, {\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3465 \, \cos \left (d x + c\right )^{10} - 108570 \, \cos \left (d x + c\right )^{8} + 482328 \, \cos \left (d x + c\right )^{6} - 574992 \, \cos \left (d x + c\right )^{4} + 98560 \, \cos \left (d x + c\right )^{2} + 32256\right )} \sin \left (d x + c\right ) + 43008}{430080 \, {\left (a^{8} d \cos \left (d x + c\right )^{12} - 32 \, a^{8} d \cos \left (d x + c\right )^{10} + 160 \, a^{8} d \cos \left (d x + c\right )^{8} - 256 \, a^{8} d \cos \left (d x + c\right )^{6} + 128 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{10} - 10 \, a^{8} d \cos \left (d x + c\right )^{8} + 24 \, a^{8} d \cos \left (d x + c\right )^{6} - 16 \, a^{8} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 186, normalized size = 0.65 \[ \frac {\frac {27720 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac {27720 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac {420 \, {\left (99 \, \sin \left (d x + c\right )^{2} - 220 \, \sin \left (d x + c\right ) + 123\right )}}{a^{8} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {81191 \, \sin \left (d x + c\right )^{10} + 858110 \, \sin \left (d x + c\right )^{9} + 4107195 \, \sin \left (d x + c\right )^{8} + 11748840 \, \sin \left (d x + c\right )^{7} + 22318590 \, \sin \left (d x + c\right )^{6} + 29583540 \, \sin \left (d x + c\right )^{5} + 27983550 \, \sin \left (d x + c\right )^{4} + 19002600 \, \sin \left (d x + c\right )^{3} + 9206235 \, \sin \left (d x + c\right )^{2} + 3108990 \, \sin \left (d x + c\right ) + 648327}{a^{8} {\left (\sin \left (d x + c\right ) + 1\right )}^{10}}}{3440640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 252, normalized size = 0.89 \[ \frac {1}{4096 a^{8} d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {11}{4096 a^{8} d \left (\sin \left (d x +c \right )-1\right )}-\frac {33 \ln \left (\sin \left (d x +c \right )-1\right )}{4096 a^{8} d}-\frac {1}{80 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{10}}-\frac {1}{48 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{9}}-\frac {3}{128 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {5}{224 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {5}{256 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {21}{1280 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {7}{512 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{256 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {45}{4096 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {55}{4096 a^{8} d \left (1+\sin \left (d x +c \right )\right )}+\frac {33 \ln \left (1+\sin \left (d x +c \right )\right )}{4096 a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 305, normalized size = 1.07 \[ -\frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{11} + 27720 \, \sin \left (d x + c\right )^{10} + 91245 \, \sin \left (d x + c\right )^{9} + 147840 \, \sin \left (d x + c\right )^{8} + 82698 \, \sin \left (d x + c\right )^{7} - 114576 \, \sin \left (d x + c\right )^{6} - 255222 \, \sin \left (d x + c\right )^{5} - 190080 \, \sin \left (d x + c\right )^{4} - 21395 \, \sin \left (d x + c\right )^{3} + 72776 \, \sin \left (d x + c\right )^{2} + 66953 \, \sin \left (d x + c\right ) + 34816\right )}}{a^{8} \sin \left (d x + c\right )^{12} + 8 \, a^{8} \sin \left (d x + c\right )^{11} + 26 \, a^{8} \sin \left (d x + c\right )^{10} + 40 \, a^{8} \sin \left (d x + c\right )^{9} + 15 \, a^{8} \sin \left (d x + c\right )^{8} - 48 \, a^{8} \sin \left (d x + c\right )^{7} - 84 \, a^{8} \sin \left (d x + c\right )^{6} - 48 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 40 \, a^{8} \sin \left (d x + c\right )^{3} + 26 \, a^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{8} \sin \left (d x + c\right ) + a^{8}} - \frac {3465 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {3465 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{430080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 290, normalized size = 1.02 \[ \frac {33\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{2048\,a^8\,d}-\frac {\frac {33\,{\sin \left (c+d\,x\right )}^{11}}{2048}+\frac {33\,{\sin \left (c+d\,x\right )}^{10}}{256}+\frac {869\,{\sin \left (c+d\,x\right )}^9}{2048}+\frac {11\,{\sin \left (c+d\,x\right )}^8}{16}+\frac {1969\,{\sin \left (c+d\,x\right )}^7}{5120}-\frac {341\,{\sin \left (c+d\,x\right )}^6}{640}-\frac {42537\,{\sin \left (c+d\,x\right )}^5}{35840}-\frac {99\,{\sin \left (c+d\,x\right )}^4}{112}-\frac {4279\,{\sin \left (c+d\,x\right )}^3}{43008}+\frac {9097\,{\sin \left (c+d\,x\right )}^2}{26880}+\frac {66953\,\sin \left (c+d\,x\right )}{215040}+\frac {17}{105}}{d\,\left (a^8\,{\sin \left (c+d\,x\right )}^{12}+8\,a^8\,{\sin \left (c+d\,x\right )}^{11}+26\,a^8\,{\sin \left (c+d\,x\right )}^{10}+40\,a^8\,{\sin \left (c+d\,x\right )}^9+15\,a^8\,{\sin \left (c+d\,x\right )}^8-48\,a^8\,{\sin \left (c+d\,x\right )}^7-84\,a^8\,{\sin \left (c+d\,x\right )}^6-48\,a^8\,{\sin \left (c+d\,x\right )}^5+15\,a^8\,{\sin \left (c+d\,x\right )}^4+40\,a^8\,{\sin \left (c+d\,x\right )}^3+26\,a^8\,{\sin \left (c+d\,x\right )}^2+8\,a^8\,\sin \left (c+d\,x\right )+a^8\right )} \]
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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