Optimal. Leaf size=114 \[ \frac {(a C+b B) \tan ^3(c+d x)}{3 d}+\frac {(a C+b B) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(3 a B+4 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.23, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3029, 2968, 3021, 2748, 3767, 3768, 3770} \[ \frac {(a C+b B) \tan ^3(c+d x)}{3 d}+\frac {(a C+b B) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(3 a B+4 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2968
Rule 3021
Rule 3029
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\int (a+b \cos (c+d x)) (B+C \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\int \left (a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (4 (b B+a C)+(3 a B+4 b C) \cos (c+d x)) \sec ^4(c+d x) \, dx\\ &=\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+(b B+a C) \int \sec ^4(c+d x) \, dx+\frac {1}{4} (3 a B+4 b C) \int \sec ^3(c+d x) \, dx\\ &=\frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 a B+4 b C) \int \sec (c+d x) \, dx-\frac {(b B+a C) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 85, normalized size = 0.75 \[ \frac {3 (3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (a C+b B) (\cos (2 (c+d x))+2) \sec (c+d x)+6 a B \sec ^2(c+d x)+9 a B+12 b C\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 136, normalized size = 1.19 \[ \frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{2} + 6 \, B a + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 304, normalized size = 2.67 \[ \frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 171, normalized size = 1.50 \[ \frac {2 a C \tan \left (d x +c \right )}{3 d}+\frac {a C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {C b \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {C b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 B b \tan \left (d x +c \right )}{3 d}+\frac {B b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 163, normalized size = 1.43 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, B a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 194, normalized size = 1.70 \[ \frac {\left (\frac {5\,B\,a}{4}-2\,B\,b-2\,C\,a+C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,B\,a}{4}+\frac {10\,B\,b}{3}+\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,B\,a}{4}-\frac {10\,B\,b}{3}-\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,B\,a}{4}+2\,B\,b+2\,C\,a+C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,B\,a}{4}+C\,b\right )}{d} \]
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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