Optimal. Leaf size=245 \[ \frac {128 \tan (c+d x)}{12155 a^8 d}-\frac {64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac {16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac {3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac {\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.40, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac {128 \tan (c+d x)}{12155 a^8 d}-\frac {64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac {24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac {3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac {\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2672
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}+\frac {9 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{17 a}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}+\frac {24 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{85 a^2}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}+\frac {168 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{1105 a^3}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}+\frac {1008 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{12155 a^4}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {112 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{2431 a^5}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {64 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{2431 a^6}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {192 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{12155 a^7}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \int \sec ^2(c+d x) \, dx}{12155 a^8}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac {128 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{12155 a^8 d}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{12155 a^8 d}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 113, normalized size = 0.46 \[ \frac {\sec (c+d x) (4862 \sin (c+d x)-6188 \sin (3 (c+d x))+1700 \sin (5 (c+d x))-119 \sin (7 (c+d x))+\sin (9 (c+d x))-7072 \cos (2 (c+d x))+3808 \cos (4 (c+d x))-544 \cos (6 (c+d x))+16 \cos (8 (c+d x)))}{24310 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 225, normalized size = 0.92 \[ \frac {1024 \, \cos \left (d x + c\right )^{8} - 10752 \, \cos \left (d x + c\right )^{6} + 29568 \, \cos \left (d x + c\right )^{4} - 27456 \, \cos \left (d x + c\right )^{2} + {\left (128 \, \cos \left (d x + c\right )^{8} - 4032 \, \cos \left (d x + c\right )^{6} + 18480 \, \cos \left (d x + c\right )^{4} - 24024 \, \cos \left (d x + c\right )^{2} + 6435\right )} \sin \left (d x + c\right ) + 5720}{12155 \, {\left (a^{8} d \cos \left (d x + c\right )^{9} - 32 \, a^{8} d \cos \left (d x + c\right )^{7} + 160 \, a^{8} d \cos \left (d x + c\right )^{5} - 256 \, a^{8} d \cos \left (d x + c\right )^{3} + 128 \, a^{8} d \cos \left (d x + c\right ) - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{7} - 10 \, a^{8} d \cos \left (d x + c\right )^{5} + 24 \, a^{8} d \cos \left (d x + c\right )^{3} - 16 \, a^{8} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.52, size = 249, normalized size = 1.02 \[ -\frac {\frac {12155}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {6211205 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 55791450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 303072770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 1091397450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 5901218466 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9405145178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 11877161010 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 9710430158 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6263238566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3172666718 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 365303990 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77883902 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10498214 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 982907}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{17}}}{3111680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 280, normalized size = 1.14 \[ \frac {-\frac {1}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {256}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {2752}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {1568}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {42800}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {5384}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {77908}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {38218}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {6847}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {10241}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {12799}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {13313}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {57083}{80 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {7937}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4351}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1793}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {511}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 740, normalized size = 3.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.04, size = 233, normalized size = 0.95 \[ \frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {519571\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}-\frac {576147\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {213707\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {183243\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {18207\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}+\frac {13855\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{16}+\frac {493\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}-\frac {237\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}+\frac {56425\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {51563\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-\frac {53191\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{2}+\frac {47003\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}+\frac {9403\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}-\frac {7703\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {355\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{2}+118\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )+\frac {\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{2}\right )}{3111680\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{17}\,\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )} \]
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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