Optimal. Leaf size=194 \[ -\frac {1}{256 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac {1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac {1}{256 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{128 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac {1}{28 a d (a \sin (c+d x)+a)^7}-\frac {1}{16 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.11, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2667, 44, 206} \[ -\frac {1}{256 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {1}{256 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac {1}{128 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{48 a^2 d (a \sin (c+d x)+a)^6}+\frac {\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac {1}{28 a d (a \sin (c+d x)+a)^7}-\frac {1}{16 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^9} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {1}{2 a (a+x)^9}+\frac {1}{4 a^2 (a+x)^8}+\frac {1}{8 a^3 (a+x)^7}+\frac {1}{16 a^4 (a+x)^6}+\frac {1}{32 a^5 (a+x)^5}+\frac {1}{64 a^6 (a+x)^4}+\frac {1}{128 a^7 (a+x)^3}+\frac {1}{256 a^8 (a+x)^2}+\frac {1}{256 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {1}{16 d (a+a \sin (c+d x))^8}-\frac {1}{28 a d (a+a \sin (c+d x))^7}-\frac {1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac {1}{80 a^3 d (a+a \sin (c+d x))^5}-\frac {1}{192 a^5 d (a+a \sin (c+d x))^3}-\frac {1}{128 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {1}{256 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {1}{256 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{256 a^7 d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac {1}{16 d (a+a \sin (c+d x))^8}-\frac {1}{28 a d (a+a \sin (c+d x))^7}-\frac {1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac {1}{80 a^3 d (a+a \sin (c+d x))^5}-\frac {1}{192 a^5 d (a+a \sin (c+d x))^3}-\frac {1}{128 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {1}{256 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {1}{256 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 122, normalized size = 0.63 \[ -\frac {105 \sin ^7(c+d x)+840 \sin ^6(c+d x)+2975 \sin ^5(c+d x)+6160 \sin ^4(c+d x)+8351 \sin ^3(c+d x)+8008 \sin ^2(c+d x)+5993 \sin (c+d x)-105 \tanh ^{-1}(\sin (c+d x)) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{16}+4096}{26880 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 374, normalized size = 1.93 \[ \frac {1680 \, \cos \left (d x + c\right )^{6} - 17360 \, \cos \left (d x + c\right )^{4} + 45696 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (\cos \left (d x + c\right )^{8} - 32 \, \cos \left (d x + c\right )^{6} + 160 \, \cos \left (d x + c\right )^{4} - 256 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 24 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 128\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (\cos \left (d x + c\right )^{8} - 32 \, \cos \left (d x + c\right )^{6} + 160 \, \cos \left (d x + c\right )^{4} - 256 \, \cos \left (d x + c\right )^{2} - 8 \, {\left (\cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 24 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 128\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 3290 \, \cos \left (d x + c\right )^{4} + 14616 \, \cos \left (d x + c\right )^{2} - 17424\right )} \sin \left (d x + c\right ) - 38208}{53760 \, {\left (a^{8} d \cos \left (d x + c\right )^{8} - 32 \, a^{8} d \cos \left (d x + c\right )^{6} + 160 \, a^{8} d \cos \left (d x + c\right )^{4} - 256 \, a^{8} d \cos \left (d x + c\right )^{2} + 128 \, a^{8} d - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 10 \, a^{8} d \cos \left (d x + c\right )^{4} + 24 \, a^{8} d \cos \left (d x + c\right )^{2} - 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 131, normalized size = 0.68 \[ \frac {\frac {840 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac {840 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} - \frac {2283 \, \sin \left (d x + c\right )^{8} + 19944 \, \sin \left (d x + c\right )^{7} + 77364 \, \sin \left (d x + c\right )^{6} + 175448 \, \sin \left (d x + c\right )^{5} + 258370 \, \sin \left (d x + c\right )^{4} + 261464 \, \sin \left (d x + c\right )^{3} + 192052 \, \sin \left (d x + c\right )^{2} + 114152 \, \sin \left (d x + c\right ) + 67819}{a^{8} {\left (\sin \left (d x + c\right ) + 1\right )}^{8}}}{430080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 180, normalized size = 0.93 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{512 a^{8} d}-\frac {1}{16 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {1}{28 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {1}{48 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {1}{80 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{128 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{192 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{256 a^{8} d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{256 a^{8} d \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{512 a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 213, normalized size = 1.10 \[ -\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 2975 \, \sin \left (d x + c\right )^{5} + 6160 \, \sin \left (d x + c\right )^{4} + 8351 \, \sin \left (d x + c\right )^{3} + 8008 \, \sin \left (d x + c\right )^{2} + 5993 \, \sin \left (d x + c\right ) + 4096\right )}}{a^{8} \sin \left (d x + c\right )^{8} + 8 \, a^{8} \sin \left (d x + c\right )^{7} + 28 \, a^{8} \sin \left (d x + c\right )^{6} + 56 \, a^{8} \sin \left (d x + c\right )^{5} + 70 \, a^{8} \sin \left (d x + c\right )^{4} + 56 \, a^{8} \sin \left (d x + c\right )^{3} + 28 \, a^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{8} \sin \left (d x + c\right ) + a^{8}} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{53760 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 198, normalized size = 1.02 \[ \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{256\,a^8\,d}-\frac {\frac {{\sin \left (c+d\,x\right )}^7}{256}+\frac {{\sin \left (c+d\,x\right )}^6}{32}+\frac {85\,{\sin \left (c+d\,x\right )}^5}{768}+\frac {11\,{\sin \left (c+d\,x\right )}^4}{48}+\frac {1193\,{\sin \left (c+d\,x\right )}^3}{3840}+\frac {143\,{\sin \left (c+d\,x\right )}^2}{480}+\frac {5993\,\sin \left (c+d\,x\right )}{26880}+\frac {16}{105}}{d\,\left (a^8\,{\sin \left (c+d\,x\right )}^8+8\,a^8\,{\sin \left (c+d\,x\right )}^7+28\,a^8\,{\sin \left (c+d\,x\right )}^6+56\,a^8\,{\sin \left (c+d\,x\right )}^5+70\,a^8\,{\sin \left (c+d\,x\right )}^4+56\,a^8\,{\sin \left (c+d\,x\right )}^3+28\,a^8\,{\sin \left (c+d\,x\right )}^2+8\,a^8\,\sin \left (c+d\,x\right )+a^8\right )} \]
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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