Optimal. Leaf size=216 \[ \frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \sin (c+d x)}{6 d}+\frac {1}{8} x \left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right )+\frac {a (4 a B+7 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {b^4 B \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.61, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4025, 4094, 4074, 4047, 8, 4045, 3770} \[ \frac {a \left (16 a^2 A b+4 a^3 B+34 a b^2 B+19 A b^3\right ) \sin (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (24 a^2 A b^2+3 a^4 A+16 a^3 b B+32 a b^3 B+8 A b^4\right )+\frac {a (4 a B+7 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {b^4 B \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4025
Rule 4045
Rule 4047
Rule 4074
Rule 4094
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (-a (7 A b+4 a B)-\left (3 a^2 A+4 A b^2+8 a b B\right ) \sec (c+d x)-4 b^2 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a (7 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (-a \left (9 a^2 A+26 A b^2+32 a b B\right )-\left (23 a^2 A b+12 A b^3+8 a^3 B+36 a b^2 B\right ) \sec (c+d x)-12 b^3 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a (7 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \cos (c+d x) \left (4 a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right )+3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \sec (c+d x)+24 b^4 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a (7 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \cos (c+d x) \left (4 a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right )+24 b^4 B \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) x+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \sin (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a (7 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\left (b^4 B\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) x+\frac {b^4 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \sin (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {a (7 A b+4 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{12 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 210, normalized size = 0.97 \[ \frac {3 a^4 A \sin (4 (c+d x))+8 a^3 (a B+4 A b) \sin (3 (c+d x))+24 a^2 \left (a^2 A+4 a b B+6 A b^2\right ) \sin (2 (c+d x))+24 a \left (3 a^3 B+12 a^2 A b+24 a b^2 B+16 A b^3\right ) \sin (c+d x)+12 (c+d x) \left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right )-96 b^4 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 b^4 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 183, normalized size = 0.85 \[ \frac {12 \, B b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, B b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} d x + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{3} + 16 \, B a^{4} + 64 \, A a^{3} b + 144 \, B a^{2} b^{2} + 96 \, A a b^{3} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 603, normalized size = 2.79 \[ \frac {24 \, B b^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, B b^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a b^{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 = 1.18, size = 319, normalized size = 1.48 \[ \frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 A \,a^{4} x}{8}+\frac {3 A \,a^{4} c}{8 d}+\frac {B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {2 a^{4} B \sin \left (d x +c \right )}{3 d}+\frac {4 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{3} b}{3 d}+\frac {8 A \,a^{3} b \sin \left (d x +c \right )}{3 d}+\frac {2 B \,a^{3} b \sin \left (d x +c \right ) \cos \left (d x +c \right )}{d}+2 B x \,a^{3} b +\frac {2 B \,a^{3} b c}{d}+\frac {3 A \,a^{2} b^{2} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{d}+3 A x \,a^{2} b^{2}+\frac {3 A \,a^{2} b^{2} c}{d}+\frac {6 a^{2} b^{2} B \sin \left (d x +c \right )}{d}+\frac {4 a A \,b^{3} \sin \left (d x +c \right )}{d}+4 B x a \,b^{3}+\frac {4 B a \,b^{3} c}{d}+A x \,b^{4}+\frac {A \,b^{4} c}{d}+\frac {B \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 215, normalized size = 1.00 \[ \frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} b + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 384 \, {\left (d x + c\right )} B a b^{3} + 96 \, {\left (d x + c\right )} A b^{4} + 48 \, B b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 576 \, B a^{2} b^{2} \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 369, normalized size = 1.71 \[ \frac {3\,B\,a^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {2\,A\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,b^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {B\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {8\,B\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,B\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {6\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {6\,A\,a^2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {4\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]
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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