Optimal. Leaf size=279 \[ \frac {128 \tan ^3(c+d x)}{12597 a^8 d}+\frac {128 \tan (c+d x)}{4199 a^8 d}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac {112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac {11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.42, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2672, 3767} \[ \frac {128 \tan ^3(c+d x)}{12597 a^8 d}+\frac {128 \tan (c+d x)}{4199 a^8 d}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac {11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac {\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}+\frac {11 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{19 a}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}+\frac {110 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{323 a^2}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}+\frac {66 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{323 a^3}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}+\frac {528 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{4199 a^4}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {336 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{4199 a^5}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {224 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{4199 a^6}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {160 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{4199 a^7}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \int \sec ^4(c+d x) \, dx}{4199 a^8}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac {128 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{4199 a^8 d}\\ &=-\frac {\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac {11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac {22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac {66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac {48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{4199 a^8 d}+\frac {128 \tan ^3(c+d x)}{12597 a^8 d}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 125, normalized size = 0.45 \[ \frac {\sec ^3(c+d x) (8398 \sin (c+d x)-5814 \sin (3 (c+d x))-2907 \sin (5 (c+d x))+1463 \sin (7 (c+d x))-117 \sin (9 (c+d x))+\sin (11 (c+d x))-10336 \cos (2 (c+d x))+2736 \cos (6 (c+d x))-512 \cos (8 (c+d x))+16 \cos (10 (c+d x)))}{50388 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 249, normalized size = 0.89 \[ \frac {2048 \, \cos \left (d x + c\right )^{10} - 21504 \, \cos \left (d x + c\right )^{8} + 59136 \, \cos \left (d x + c\right )^{6} - 54912 \, \cos \left (d x + c\right )^{4} + 11440 \, \cos \left (d x + c\right )^{2} + {\left (256 \, \cos \left (d x + c\right )^{10} - 8064 \, \cos \left (d x + c\right )^{8} + 36960 \, \cos \left (d x + c\right )^{6} - 48048 \, \cos \left (d x + c\right )^{4} + 12870 \, \cos \left (d x + c\right )^{2} + 2431\right )} \sin \left (d x + c\right ) + 1768}{12597 \, {\left (a^{8} d \cos \left (d x + c\right )^{11} - 32 \, a^{8} d \cos \left (d x + c\right )^{9} + 160 \, a^{8} d \cos \left (d x + c\right )^{7} - 256 \, a^{8} d \cos \left (d x + c\right )^{5} + 128 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{9} - 10 \, a^{8} d \cos \left (d x + c\right )^{7} + 24 \, a^{8} d \cos \left (d x + c\right )^{5} - 16 \, a^{8} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 301, normalized size = 1.08 \[ -\frac {\frac {4199 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {12823746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} + 140368371 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 879644311 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 3693272440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 27403194676 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 51919375300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 79183835016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 99750226290 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 82860874122 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 56110430792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 13462452660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4616712644 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1197851960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 27911475 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2143959}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{19}}}{6449664 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 340, normalized size = 1.22 \[ \frac {-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {256}{19 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{19}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{18}}-\frac {10496}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {1984}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {14192}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {8856}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {175016}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {50936}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {18011}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {32417}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {12430}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {32525}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {72425}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {204605}{96 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {26871}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2177}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {54229}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {7181}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {509}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 866, normalized size = 3.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.23, size = 277, normalized size = 0.99 \[ \frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {896971\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {1062347\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {40375\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {40375\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {412471\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}-\frac {324919\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}-\frac {11305\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {7209\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}+\frac {765\,\cos \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{128}-\frac {253\,\cos \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{128}+\frac {65033\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {56635\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-6271\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+\frac {9635\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}-\frac {9635\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}+\frac {16363\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}+\frac {10537\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{8}-\frac {7611\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{8}-\frac {485\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{8}+\frac {251\,\sin \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{8}+\frac {\sin \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{4}\right )}{12899328\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{19}\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \]
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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