Optimal. Leaf size=162 \[ \frac {5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac {(12 A+20 B+15 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{24 d}+\frac {1}{8} a^3 x (28 A+20 B+15 C)+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(4 B+3 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 a d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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Rubi [A] time = 0.48, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac {(12 A+20 B+15 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{24 d}+\frac {1}{8} a^3 x (28 A+20 B+15 C)+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(4 B+3 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 a d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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))^3 \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))^3 \sin (c+d x)}{4 d}+\frac {\int (a+a \cos (c+d x))^3 (4 a A+a (4 B+3 C) \cos (c+d x)) \sec (c+d x) \, dx}{4 a}\\ &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {\int (a+a \cos (c+d x))^2 \left (12 a^2 A+a^2 (12 A+20 B+15 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int (a+a \cos (c+d x)) \left (24 a^3 A+15 a^3 (4 A+4 B+3 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \left (24 a^4 A+\left (24 a^4 A+15 a^4 (4 A+4 B+3 C)\right ) \cos (c+d x)+15 a^4 (4 A+4 B+3 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \left (24 a^4 A+3 a^4 (28 A+20 B+15 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {1}{8} a^3 (28 A+20 B+15 C) x+\frac {5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^3 (28 A+20 B+15 C) x+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 147, normalized size = 0.91 \[ \frac {a^3 \left (24 (12 A+15 B+13 C) \sin (c+d x)+24 (A+3 B+4 C) \sin (2 (c+d x))-96 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+336 A d x+8 B \sin (3 (c+d x))+240 B d x+24 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+180 C d x\right )}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 131, normalized size = 0.81 \[ \frac {3 \, {\left (28 \, A + 20 \, B + 15 \, C\right )} a^{3} d x + 12 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 12 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (9 \, A + 11 \, B + 9 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 286, normalized size = 1.77 \[ \frac {24 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (28 \, A a^{3} + 20 \, B a^{3} + 15 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (60 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 204 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 220 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 228 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 292 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 219 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 132 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 147 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 251, normalized size = 1.55 \[ \frac {A \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 A x \,a^{3}}{2}+\frac {7 A \,a^{3} c}{2 d}+\frac {B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{3 d}+\frac {11 a^{3} B \sin \left (d x +c \right )}{3 d}+\frac {C \,a^{3} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {15 C \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {15 a^{3} C x}{8}+\frac {15 C \,a^{3} c}{8 d}+\frac {3 a^{3} A \sin \left (d x +c \right )}{d}+\frac {3 a^{3} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {5 a^{3} B x}{2}+\frac {5 a^{3} B c}{2 d}+\frac {C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{d}+\frac {3 a^{3} C \sin \left (d x +c \right )}{d}+\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 233, normalized size = 1.44 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 288 \, {\left (d x + c\right )} A a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 96 \, {\left (d x + c\right )} B a^{3} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 96 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 288 \, A a^{3} \sin \left (d x + c\right ) + 288 \, B a^{3} \sin \left (d x + c\right ) + 96 \, C a^{3} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 242, normalized size = 1.49 \[ \frac {7\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+5\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+\frac {15\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}+\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12}+C\,a^3\,\sin \left (2\,c+2\,d\,x\right )+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32}+3\,A\,a^3\,\sin \left (c+d\,x\right )+\frac {15\,B\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {13\,C\,a^3\,\sin \left (c+d\,x\right )}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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