Optimal. Leaf size=225 \[ \frac {a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac {a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(28 A+35 B+32 C) \tan (c+d x) \sec (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+a^4 C x+\frac {(28 A+35 B+20 C) \tan (c+d x) \sec ^2(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{60 d}+\frac {a (4 A+5 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.69, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3043, 2975, 2968, 3021, 2735, 3770} \[ \frac {a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac {a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(28 A+35 B+20 C) \tan (c+d x) \sec ^2(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{60 d}+\frac {(28 A+35 B+32 C) \tan (c+d x) \sec (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+a^4 C x+\frac {a (4 A+5 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 3021
Rule 3043
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^4 (a (4 A+5 B)+5 a C \cos (c+d x)) \sec ^5(c+d x) \, dx}{5 a}\\ &=\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^3 \left (a^2 (28 A+35 B+20 C)+20 a^2 C \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{20 a}\\ &=\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^2 \left (5 a^3 (28 A+35 B+32 C)+60 a^3 C \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{60 a}\\ &=\frac {(28 A+35 B+32 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x)) \left (15 a^4 (28 A+35 B+40 C)+120 a^4 C \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{120 a}\\ &=\frac {(28 A+35 B+32 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \left (15 a^5 (28 A+35 B+40 C)+\left (120 a^5 C+15 a^5 (28 A+35 B+40 C)\right ) \cos (c+d x)+120 a^5 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx}{120 a}\\ &=\frac {a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \left (15 a^5 (28 A+35 B+48 C)+120 a^5 C \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=a^4 C x+\frac {a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{8} \left (a^4 (28 A+35 B+48 C)\right ) \int \sec (c+d x) \, dx\\ &=a^4 C x+\frac {a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(28 A+35 B+20 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {a (4 A+5 B) (a+a \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] time = 6.22, size = 971, normalized size = 4.32 \[ \frac {C (c+d x) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{16 d}+\frac {(-28 A-35 B-48 C) (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d}+\frac {(28 A+35 B+48 C) (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d}+\frac {A (\cos (c+d x) a+a)^4 \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{320 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {(\cos (c+d x) a+a)^4 \left (139 A \sin \left (\frac {1}{2} (c+d x)\right )+80 B \sin \left (\frac {1}{2} (c+d x)\right )+20 C \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {(\cos (c+d x) a+a)^4 \left (139 A \sin \left (\frac {1}{2} (c+d x)\right )+80 B \sin \left (\frac {1}{2} (c+d x)\right )+20 C \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {(\cos (c+d x) a+a)^4 \left (83 A \sin \left (\frac {1}{2} (c+d x)\right )+100 B \sin \left (\frac {1}{2} (c+d x)\right )+100 C \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{240 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {(\cos (c+d x) a+a)^4 \left (83 A \sin \left (\frac {1}{2} (c+d x)\right )+100 B \sin \left (\frac {1}{2} (c+d x)\right )+100 C \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{240 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {(559 A+485 B+260 C) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{3840 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {(-559 A-485 B-260 C) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{3840 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {(22 A+5 B) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{1280 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {(-22 A-5 B) (\cos (c+d x) a+a)^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{1280 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {A (\cos (c+d x) a+a)^4 \sin \left (\frac {1}{2} (c+d x)\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right )}{320 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 196, normalized size = 0.87 \[ \frac {240 \, C a^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (28 \, A + 35 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (28 \, A + 35 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (83 \, A + 100 \, B + 100 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, A + 27 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (34 \, A + 20 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 352, normalized size = 1.56 \[ \frac {120 \, {\left (d x + c\right )} C a^{4} + 15 \, {\left (28 \, A a^{4} + 35 \, B a^{4} + 48 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (28 \, A a^{4} + 35 \, B a^{4} + 48 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (420 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 600 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1960 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3584 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3160 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3680 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1500 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1080 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 331, normalized size = 1.47 \[ \frac {83 A \,a^{4} \tan \left (d x +c \right )}{15 d}+\frac {35 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+a^{4} C x +\frac {a^{4} C c}{d}+\frac {7 A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {7 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {20 a^{4} B \tan \left (d x +c \right )}{3 d}+\frac {6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {34 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d}+\frac {27 a^{4} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {20 a^{4} C \tan \left (d x +c \right )}{3 d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {4 a^{4} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {2 a^{4} C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} B \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {a^{4} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 496, normalized size = 2.20 \[ \frac {16 \, {\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} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 240 \, {\left (d x + c\right )} C a^{4} - 60 \, 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 )} - 15 \, B 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 )} - 240 \, A 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 )} - 360 \, B 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 )} - 240 \, 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 )} + 120 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{4} \tan \left (d x + c\right ) + 960 \, B a^{4} \tan \left (d x + c\right ) + 1440 \, C a^{4} \tan \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.80, size = 995, normalized size = 4.42 \[ \text {result too large to display} \]
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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