Optimal. Leaf size=147 \[ \frac {5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac {(4 A+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{8 d}+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^3 x (28 A+15 C)+\frac {C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{4 a d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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Rubi [A] time = 0.44, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3046, 2976, 2968, 3023, 2735, 3770} \[ \frac {5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac {(4 A+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{8 d}+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^3 x (28 A+15 C)+\frac {C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{4 a d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3046
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (A+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+3 a 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 {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {\int (a+a \cos (c+d x))^2 \left (12 a^2 A+3 a^2 (4 A+5 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 {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{8 d}+\frac {\int (a+a \cos (c+d x)) \left (24 a^3 A+15 a^3 (4 A+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 {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{8 d}+\frac {\int \left (24 a^4 A+\left (24 a^4 A+15 a^4 (4 A+3 C)\right ) \cos (c+d x)+15 a^4 (4 A+3 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{8 d}+\frac {\int \left (24 a^4 A+3 a^4 (28 A+15 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {1}{8} a^3 (28 A+15 C) x+\frac {5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{8 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^3 (28 A+15 C) x+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{4 a d}+\frac {(4 A+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 124, normalized size = 0.84 \[ \frac {a^3 \left (8 (12 A+13 C) \sin (c+d x)+8 (A+4 C) \sin (2 (c+d x))-32 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+112 A d x+8 C \sin (3 (c+d x))+C \sin (4 (c+d x))+60 C d x\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.48, size = 112, normalized size = 0.76 \[ \frac {{\left (28 \, A + 15 \, C\right )} a^{3} d x + 4 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \, C a^{3} \cos \left (d x + c\right )^{2} + {\left (4 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right ) + 24 \, {\left (A + C\right )} a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 213, normalized size = 1.45 \[ \frac {8 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (28 \, A a^{3} + 15 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (20 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 68 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 55 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 28 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 49 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 175, normalized size = 1.19 \[ \frac {A \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 A x \,a^{3}}{2}+\frac {7 A \,a^{3} c}{2 d}+\frac {C \,a^{3} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {15 C \,a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {15 a^{3} C x}{8}+\frac {15 C \,a^{3} c}{8 d}+\frac {3 a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{d}+\frac {3 a^{3} C \sin \left (d x +c \right )}{d}+\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 163, normalized size = 1.11 \[ \frac {8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 96 \, {\left (d x + c\right )} A a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 32 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, A a^{3} \sin \left (d x + c\right ) + 32 \, C a^{3} \sin \left (d x + c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 195, normalized size = 1.33 \[ \frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {13\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {7\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {15\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
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 \[ a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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