Optimal. Leaf size=195 \[ \frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (48 A+35 B+28 C)+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(20 A+35 B+28 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{60 d}+\frac {a (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.60, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3045, 2976, 2968, 3023, 2735, 3770} \[ \frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {(20 A+35 B+28 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{60 d}+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (48 A+35 B+28 C)+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3045
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 (c+d x) \, dx &=\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^4 (5 a A+a (5 B+4 C) \cos (c+d x)) \sec (c+d x) \, dx}{5 a}\\ &=\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^3 \left (20 a^2 A+a^2 (20 A+35 B+28 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{20 a}\\ &=\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {\int (a+a \cos (c+d x))^2 \left (60 a^3 A+5 a^3 (32 A+35 B+28 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{60 a}\\ &=\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int (a+a \cos (c+d x)) \left (120 a^4 A+15 a^4 (40 A+35 B+28 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \left (120 a^5 A+\left (120 a^5 A+15 a^5 (40 A+35 B+28 C)\right ) \cos (c+d x)+15 a^5 (40 A+35 B+28 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \left (120 a^5 A+15 a^5 (48 A+35 B+28 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {1}{8} a^4 (48 A+35 B+28 C) x+\frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (48 A+35 B+28 C) x+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 182, normalized size = 0.93 \[ \frac {a^4 \left (60 (54 A+56 B+49 C) \sin (c+d x)+120 (4 A+7 B+8 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))-480 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+2880 A d x+160 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+2100 B d x+290 C \sin (3 (c+d x))+60 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+1680 C d x\right )}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 154, normalized size = 0.79 \[ \frac {15 \, {\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} d x + 60 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, C a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 20 \, B + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (16 \, A + 27 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (100 \, A + 100 \, B + 83 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 337, normalized size = 1.73 \[ \frac {120 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (600 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1960 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3584 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3680 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3160 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1080 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1500 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 320, normalized size = 1.64 \[ \frac {7 a^{4} C x}{2}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {7 a^{4} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 A \,a^{4} \sin \left (d x +c \right )}{3 d}+\frac {83 a^{4} C \sin \left (d x +c \right )}{15 d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {34 a^{4} C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}+\frac {4 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 a^{4} B \sin \left (d x +c \right )}{3 d}+\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {35 a^{4} B c}{8 d}+\frac {6 A \,a^{4} c}{d}+\frac {7 a^{4} C c}{2 d}+\frac {35 a^{4} B x}{8}+\frac {a^{4} B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {27 a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+6 A \,a^{4} x +\frac {2 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 325, normalized size = 1.67 \[ -\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1920 \, {\left (d x + c\right )} A a^{4} + 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \, {\left (d x + c\right )} B a^{4} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 2880 \, A a^{4} \sin \left (d x + c\right ) - 1920 \, B a^{4} \sin \left (d x + c\right ) - 480 \, C a^{4} \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.52, size = 1151, normalized size = 5.90 \[ \text {result too large to display} \]
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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