Optimal. Leaf size=135 \[ \frac {a^2 (4 A+5 B) \sin (c+d x)}{3 d}+\frac {a^2 (5 A+4 B) \sin (c+d x) \cos ^2(c+d x)}{12 d}+\frac {a^2 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^2 x (7 A+8 B)+\frac {A \sin (c+d x) \cos ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{4 d} \]
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Rubi [A] time = 0.23, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4017, 3996, 3787, 2635, 8, 2637} \[ \frac {a^2 (4 A+5 B) \sin (c+d x)}{3 d}+\frac {a^2 (5 A+4 B) \sin (c+d x) \cos ^2(c+d x)}{12 d}+\frac {a^2 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^2 x (7 A+8 B)+\frac {A \sin (c+d x) \cos ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 3787
Rule 3996
Rule 4017
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+a \sec (c+d x)) (a (5 A+4 B)+2 a (A+2 B) \sec (c+d x)) \, dx\\ &=\frac {a^2 (5 A+4 B) \cos ^2(c+d x) \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) \left (-3 a^2 (7 A+8 B)-4 a^2 (4 A+5 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^2 (5 A+4 B) \cos ^2(c+d x) \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{4 d}+\frac {1}{3} \left (a^2 (4 A+5 B)\right ) \int \cos (c+d x) \, dx+\frac {1}{4} \left (a^2 (7 A+8 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a^2 (4 A+5 B) \sin (c+d x)}{3 d}+\frac {a^2 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (5 A+4 B) \cos ^2(c+d x) \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{4 d}+\frac {1}{8} \left (a^2 (7 A+8 B)\right ) \int 1 \, dx\\ &=\frac {1}{8} a^2 (7 A+8 B) x+\frac {a^2 (4 A+5 B) \sin (c+d x)}{3 d}+\frac {a^2 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (5 A+4 B) \cos ^2(c+d x) \sin (c+d x)}{12 d}+\frac {A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 86, normalized size = 0.64 \[ \frac {a^2 (24 (6 A+7 B) \sin (c+d x)+48 (A+B) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+84 A c+84 A d x+8 B \sin (3 (c+d x))+96 B d x)}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 90, normalized size = 0.67 \[ \frac {3 \, {\left (7 \, A + 8 \, B\right )} a^{2} d x + {\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (7 \, A + 8 \, B\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (4 \, A + 5 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 176, normalized size = 1.30 \[ \frac {3 \, {\left (7 \, A a^{2} + 8 \, B a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (21 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 77 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 88 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 83 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 136 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.28, size = 154, normalized size = 1.14 \[ \frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 144, normalized size = 1.07 \[ -\frac {64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 134, normalized size = 0.99 \[ \frac {7\,A\,a^2\,x}{8}+B\,a^2\,x+\frac {3\,A\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {A\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
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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