Optimal. Leaf size=263 \[ \frac {a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac {a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^4 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{210 d}+\frac {a^4 (11 A+14 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{210 d}+\frac {(8 A+7 C) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{35 d}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac {2 a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{21 d} \]
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Rubi [A] time = 0.80, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3044, 2975, 2968, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac {a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac {a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^4 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{210 d}+\frac {a^4 (11 A+14 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac {(8 A+7 C) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{35 d}+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{210 d}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac {2 a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{21 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2968
Rule 2975
Rule 3021
Rule 3044
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x))^4 (4 a A+a (2 A+7 C) \cos (c+d x)) \sec ^7(c+d x) \, dx}{7 a}\\ &=\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x))^3 \left (6 a^2 (8 A+7 C)+2 a^2 (10 A+21 C) \cos (c+d x)\right ) \sec ^6(c+d x) \, dx}{42 a}\\ &=\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x))^2 \left (4 a^3 (109 A+126 C)+98 a^3 (2 A+3 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{210 a}\\ &=\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x)) \left (12 a^4 (247 A+308 C)+24 a^4 (69 A+91 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{840 a}\\ &=\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int \left (12 a^5 (247 A+308 C)+\left (24 a^5 (69 A+91 C)+12 a^5 (247 A+308 C)\right ) \cos (c+d x)+24 a^5 (69 A+91 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{840 a}\\ &=\frac {a^4 (247 A+308 C) \sec ^2(c+d x) \tan (c+d x)}{210 d}+\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {\int \left (1260 a^5 (11 A+14 C)+24 a^5 (454 A+581 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{2520 a}\\ &=\frac {a^4 (247 A+308 C) \sec ^2(c+d x) \tan (c+d x)}{210 d}+\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {1}{2} \left (a^4 (11 A+14 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{105} \left (a^4 (454 A+581 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {a^4 (11 A+14 C) \sec (c+d x) \tan (c+d x)}{4 d}+\frac {a^4 (247 A+308 C) \sec ^2(c+d x) \tan (c+d x)}{210 d}+\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {1}{4} \left (a^4 (11 A+14 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^4 (454 A+581 C)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 d}\\ &=\frac {a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac {a^4 (11 A+14 C) \sec (c+d x) \tan (c+d x)}{4 d}+\frac {a^4 (247 A+308 C) \sec ^2(c+d x) \tan (c+d x)}{210 d}+\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 3.17, size = 390, normalized size = 1.48 \[ -\frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^7(c+d x) \left (6720 (11 A+14 C) \cos ^7(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) (-140 (122 A+217 C) \sin (2 c+d x)+16415 A \sin (c+2 d x)+16415 A \sin (3 c+2 d x)+37296 A \sin (2 c+3 d x)-840 A \sin (4 c+3 d x)+7700 A \sin (3 c+4 d x)+7700 A \sin (5 c+4 d x)+12712 A \sin (4 c+5 d x)+1155 A \sin (5 c+6 d x)+1155 A \sin (7 c+6 d x)+1816 A \sin (6 c+7 d x)+560 (83 A+91 C) \sin (d x)+10710 C \sin (c+2 d x)+10710 C \sin (3 c+2 d x)+41244 C \sin (2 c+3 d x)-7560 C \sin (4 c+3 d x)+7560 C \sin (3 c+4 d x)+7560 C \sin (5 c+4 d x)+15848 C \sin (4 c+5 d x)-420 C \sin (6 c+5 d x)+1470 C \sin (5 c+6 d x)+1470 C \sin (7 c+6 d x)+2324 C \sin (6 c+7 d x))\right )}{430080 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 201, normalized size = 0.76 \[ \frac {105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (454 \, A + 581 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 4 \, {\left (227 \, A + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (11 \, A + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 12 \, {\left (48 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, A a^{4} \cos \left (d x + c\right ) + 60 \, A a^{4}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.79, size = 314, normalized size = 1.19 \[ \frac {105 \, {\left (11 \, A a^{4} + 14 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (11 \, A a^{4} + 14 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1155 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1470 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 7700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 9800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 21791 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 27734 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 33792 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 43008 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 31521 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 39914 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21560 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5565 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5250 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 303, normalized size = 1.15 \[ \frac {454 A \,a^{4} \tan \left (d x +c \right )}{105 d}+\frac {227 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{105 d}+\frac {83 a^{4} C \tan \left (d x +c \right )}{15 d}+\frac {11 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{6 d}+\frac {11 A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{4 d}+\frac {11 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {7 a^{4} C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {7 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {48 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{35 d}+\frac {34 a^{4} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d}+\frac {2 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{3 d}+\frac {a^{4} C \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{6}\left (d x +c \right )\right )}{7 d}+\frac {a^{4} C \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 462, normalized size = 1.76 \[ \frac {24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{4} + 336 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 56 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, A a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 840 \, C a^{4} \tan \left (d x + c\right )}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 301, normalized size = 1.14 \[ \frac {a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (11\,A+14\,C\right )}{2\,d}-\frac {\left (\frac {11\,A\,a^4}{2}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {110\,A\,a^4}{3}-\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {5632\,A\,a^4}{35}-\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-70\,A\,a^4-\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+25\,C\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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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