Optimal. Leaf size=165 \[ \frac {\tan ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac {\tan (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac {(3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac {(a B+A b) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {a A \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.26, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3031, 3021, 2748, 3767, 3768, 3770} \[ \frac {\tan ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac {\tan (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac {(3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac {(a B+A b) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {a A \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 3021
Rule 3031
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac {a A \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{5} \int \left (-5 (A b+a B)-(4 a A+5 b B+5 a C) \cos (c+d x)-5 b C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {(A b+a B) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a A \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{20} \int (-4 (4 a A+5 b B+5 a C)-5 (3 A b+3 a B+4 b C) \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac {(A b+a B) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a A \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{5} (-4 a A-5 b B-5 a C) \int \sec ^4(c+d x) \, dx-\frac {1}{4} (-3 A b-3 a B-4 b C) \int \sec ^3(c+d x) \, dx\\ &=\frac {(3 A b+3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a A \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{8} (-3 A b-3 a B-4 b C) \int \sec (c+d x) \, dx-\frac {(4 a A+5 b B+5 a C) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {(3 A b+3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(4 a A+5 b B+5 a C) \tan (c+d x)}{5 d}+\frac {(3 A b+3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a A \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {(4 a A+5 b B+5 a C) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 1.37, size = 123, normalized size = 0.75 \[ \frac {15 (3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 \left (5 \tan ^2(c+d x) (a (2 A+C)+b B)+15 (a (A+C)+b B)+3 a A \tan ^4(c+d x)\right )+15 \sec (c+d x) (3 a B+3 A b+4 b C)+30 (a B+A b) \sec ^3(c+d x)\right )}{120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 182, normalized size = 1.10 \[ \frac {15 \, {\left (3 \, B a + {\left (3 \, A + 4 \, C\right )} b\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (3 \, B a + {\left (3 \, A + 4 \, C\right )} b\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left ({\left (4 \, A + 5 \, C\right )} a + 5 \, B b\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (3 \, B a + {\left (3 \, A + 4 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a + 5 \, B b\right )} \cos \left (d x + c\right )^{2} + 24 \, A a + 30 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 473, normalized size = 2.87 \[ \frac {15 \, {\left (3 \, B a + 3 \, A b + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (3 \, B a + 3 \, A b + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (120 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 160 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 320 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 320 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 400 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 400 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C b \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 )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 287, normalized size = 1.74 \[ \frac {8 a A \tan \left (d x +c \right )}{15 d}+\frac {a A \left (\sec ^{4}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{5 d}+\frac {4 a A \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{15 d}+\frac {a B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 a C \tan \left (d x +c \right )}{3 d}+\frac {a C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {A b \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 A b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 B b \tan \left (d x +c \right )}{3 d}+\frac {B b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C b \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 266, normalized size = 1.61 \[ \frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 15 \, B a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, A b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, C b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.30, size = 302, normalized size = 1.83 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,A\,b}{8}+\frac {3\,B\,a}{8}+\frac {C\,b}{2}\right )}{\frac {3\,A\,b}{2}+\frac {3\,B\,a}{2}+2\,C\,b}\right )\,\left (\frac {3\,A\,b}{4}+\frac {3\,B\,a}{4}+C\,b\right )}{d}-\frac {\left (2\,A\,a-\frac {5\,A\,b}{4}-\frac {5\,B\,a}{4}+2\,B\,b+2\,C\,a-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b}{2}-\frac {8\,A\,a}{3}+\frac {B\,a}{2}-\frac {16\,B\,b}{3}-\frac {16\,C\,a}{3}+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,A\,a}{15}+\frac {20\,B\,b}{3}+\frac {20\,C\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {8\,A\,a}{3}-\frac {A\,b}{2}-\frac {B\,a}{2}-\frac {16\,B\,b}{3}-\frac {16\,C\,a}{3}-2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a+\frac {5\,A\,b}{4}+\frac {5\,B\,a}{4}+2\,B\,b+2\,C\,a+C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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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