Optimal. Leaf size=196 \[ \frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^4 x (52 A+48 B+35 C)-\frac {(12 A-4 B-7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}-\frac {a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d} \]
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Rubi [A] time = 0.68, antiderivative size = 196, 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, 2976, 2968, 3023, 2735, 3770} \[ \frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {(12 A-4 B-7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}-\frac {(12 A-32 B-35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^4 x (52 A+48 B+35 C)-\frac {a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
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 ^2(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^4 (a (4 A+B)-a (4 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^3 \left (4 a^2 (4 A+B)-a^2 (12 A-4 B-7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{4 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^2 \left (12 a^3 (4 A+B)-a^3 (12 A-32 B-35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x)) \left (24 a^4 (4 A+B)+15 a^4 (4 A+8 B+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (24 a^5 (4 A+B)+\left (24 a^5 (4 A+B)+15 a^5 (4 A+8 B+7 C)\right ) \cos (c+d x)+15 a^5 (4 A+8 B+7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (24 a^5 (4 A+B)+3 a^5 (52 A+48 B+35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {1}{8} a^4 (52 A+48 B+35 C) x+\frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\left (a^4 (4 A+B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (52 A+48 B+35 C) x+\frac {a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-4 B-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-32 B-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 1.91, size = 246, normalized size = 1.26 \[ \frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (12 (52 A+48 B+35 C) (c+d x)+24 (16 A+27 B+28 C) \sin (c+d x)+24 (A+4 B+7 C) \sin (2 (c+d x))-96 (4 A+B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 (4 A+B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {96 A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {96 A \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+8 (B+4 C) \sin (3 (c+d x))+3 C \sin (4 (c+d x))\right )}{1536 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 179, normalized size = 0.91 \[ \frac {3 \, {\left (52 \, A + 48 \, B + 35 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 12 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 8 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 16 \, B + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (3 \, A + 5 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.55, size = 332, normalized size = 1.69 \[ -\frac {\frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (52 \, A a^{4} + 48 \, B a^{4} + 35 \, C a^{4}\right )} {\left (d x + c\right )} - 24 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 24 \, {\left (4 \, A a^{4} + B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (84 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 276 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 424 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 289, normalized size = 1.47 \[ \frac {35 a^{4} C x}{8}+\frac {B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 a^{4} B \sin \left (d x +c \right )}{3 d}+\frac {4 a^{4} C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{3 d}+\frac {20 a^{4} C \sin \left (d x +c \right )}{3 d}+\frac {A \,a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 A \,a^{4} \sin \left (d x +c \right )}{d}+\frac {4 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{4} B c}{d}+\frac {13 A \,a^{4} c}{2 d}+\frac {2 a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {35 a^{4} C c}{8 d}+\frac {A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {27 a^{4} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+6 a^{4} B x +\frac {13 A \,a^{4} x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 290, normalized size = 1.48 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 576 \, {\left (d x + c\right )} A a^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 384 \, {\left (d x + c\right )} B a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 96 \, {\left (d x + c\right )} C a^{4} + 192 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, A a^{4} \sin \left (d x + c\right ) + 576 \, B a^{4} \sin \left (d x + c\right ) + 384 \, C a^{4} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 1244, normalized size = 6.35 \[ \text {result too large to display} \]
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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