Optimal. Leaf size=195 \[ a^3 x (a B+4 A b)-\frac {b^2 \left (6 a^2 A-8 a b B-3 A b^2\right ) \tan (c+d x) \sec (c+d x)}{6 d}-\frac {b \left (6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right ) \tan (c+d x)}{3 d}+\frac {b \left (8 a^3 B+12 a^2 A b+4 a b^2 B+A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b (3 a A-b B) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {a A \sin (c+d x) (a+b \sec (c+d x))^3}{d} \]
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Rubi [A] time = 0.37, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4025, 4056, 4048, 3770, 3767, 8} \[ -\frac {b \left (6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right ) \tan (c+d x)}{3 d}+\frac {b \left (12 a^2 A b+8 a^3 B+4 a b^2 B+A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^2 \left (6 a^2 A-8 a b B-3 A b^2\right ) \tan (c+d x) \sec (c+d x)}{6 d}+a^3 x (a B+4 A b)-\frac {b (3 a A-b B) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {a A \sin (c+d x) (a+b \sec (c+d x))^3}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 4025
Rule 4048
Rule 4056
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\int (a+b \sec (c+d x))^2 \left (-a (4 A b+a B)-b (A b+2 a B) \sec (c+d x)+b (3 a A-b B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b (3 a A-b B) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {1}{3} \int (a+b \sec (c+d x)) \left (-3 a^2 (4 A b+a B)-b \left (9 a A b+9 a^2 B+2 b^2 B\right ) \sec (c+d x)+b \left (6 a^2 A-3 A b^2-8 a b B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 \left (6 a^2 A-3 A b^2-8 a b B\right ) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 a A-b B) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-6 a^3 (4 A b+a B)-3 b \left (12 a^2 A b+A b^3+8 a^3 B+4 a b^2 B\right ) \sec (c+d x)+2 b \left (6 a^3 A-12 a A b^2-17 a^2 b B-2 b^3 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^3 (4 A b+a B) x+\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 \left (6 a^2 A-3 A b^2-8 a b B\right ) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 a A-b B) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{2} \left (b \left (12 a^2 A b+A b^3+8 a^3 B+4 a b^2 B\right )\right ) \int \sec (c+d x) \, dx-\frac {1}{3} \left (b \left (6 a^3 A-12 a A b^2-17 a^2 b B-2 b^3 B\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=a^3 (4 A b+a B) x+\frac {b \left (12 a^2 A b+A b^3+8 a^3 B+4 a b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b^2 \left (6 a^2 A-3 A b^2-8 a b B\right ) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 a A-b B) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {\left (b \left (6 a^3 A-12 a A b^2-17 a^2 b B-2 b^3 B\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=a^3 (4 A b+a B) x+\frac {b \left (12 a^2 A b+A b^3+8 a^3 B+4 a b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b \left (6 a^3 A-12 a A b^2-17 a^2 b B-2 b^3 B\right ) \tan (c+d x)}{3 d}-\frac {b^2 \left (6 a^2 A-3 A b^2-8 a b B\right ) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 a A-b B) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.29, size = 1051, normalized size = 5.39 \[ \frac {\left (-A b^4-4 a B b^3-12 a^2 A b^2-8 a^3 B b\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{2 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac {\left (A b^4+4 a B b^3+12 a^2 A b^2+8 a^3 B b\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{2 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac {a^3 (4 A b+a B) (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac {b^4 B (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^5(c+d x)}{6 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left (B \sin \left (\frac {1}{2} (c+d x)\right ) b^4+6 a A \sin \left (\frac {1}{2} (c+d x)\right ) b^3+9 a^2 B \sin \left (\frac {1}{2} (c+d x)\right ) b^2\right ) \cos ^5(c+d x)}{3 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left (B \sin \left (\frac {1}{2} (c+d x)\right ) b^4+6 a A \sin \left (\frac {1}{2} (c+d x)\right ) b^3+9 a^2 B \sin \left (\frac {1}{2} (c+d x)\right ) b^2\right ) \cos ^5(c+d x)}{3 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a^4 A (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin (c+d x) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac {\left (3 A b^4+B b^4+12 a B b^3\right ) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{12 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\left (-3 A b^4-B b^4-12 a B b^3\right ) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{12 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^4 B (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^5(c+d x)}{6 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 219, normalized size = 1.12 \[ \frac {12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, B a^{3} b + 12 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{3} b + 12 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{4} \cos \left (d x + c\right )^{3} + 2 \, B b^{4} + 4 \, {\left (9 \, B a^{2} b^{2} + 6 \, A a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 387, normalized size = 1.98 \[ \frac {\frac {12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} {\left (d x + c\right )} + 3 \, {\left (8 \, B a^{3} b + 12 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, B a^{3} b + 12 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (36 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.48, size = 262, normalized size = 1.34 \[ \frac {A \,a^{4} \sin \left (d x +c \right )}{d}+a^{4} B x +\frac {a^{4} B c}{d}+4 A \,a^{3} b x +\frac {4 A \,a^{3} b c}{d}+\frac {4 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{2} b^{2} B \tan \left (d x +c \right )}{d}+\frac {4 a A \,b^{3} \tan \left (d x +c \right )}{d}+\frac {2 B a \,b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {2 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,b^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 B \,b^{4} \tan \left (d x +c \right )}{3 d}+\frac {B \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 245, normalized size = 1.26 \[ \frac {12 \, {\left (d x + c\right )} B a^{4} + 48 \, {\left (d x + c\right )} A a^{3} b + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{4} - 12 \, B a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, A b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} \sin \left (d x + c\right ) + 72 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 48 \, A a b^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.90, size = 636, normalized size = 3.26 \[ \frac {\frac {A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {B\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {B\,b^4\,\sin \left (c+d\,x\right )}{2}+A\,a\,b^3\,\sin \left (c+d\,x\right )+\frac {3\,B\,a^4\,\cos \left (c+d\,x\right )\,\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}-\frac {A\,b^4\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4}+A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )+B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )+\frac {3\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2}+\frac {B\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {A\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{4}+\frac {3\,B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2}+2\,A\,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 )\,\cos \left (3\,c+3\,d\,x\right )-A\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}-B\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-B\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,2{}\mathrm {i}-A\,a^2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}+6\,A\,a^3\,b\,\cos \left (c+d\,x\right )\,\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 )-B\,a\,b^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}-B\,a^3\,b\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
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 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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