Optimal. Leaf size=106 \[ \frac {a (B+C) \tan ^3(c+d x)}{3 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.22, antiderivative size = 106, 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} \[ \frac {a (B+C) \tan ^3(c+d x)}{3 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d} \]
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+a \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+a \cos (c+d x)) (B+C \cos (c+d x)) \sec ^5(c+d x) \, dx\\ &=\int \left (a B+(a B+a C) \cos (c+d x)+a 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 a (B+C)+a (3 B+4 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}+(a (B+C)) \int \sec ^4(c+d x) \, dx+\frac {1}{4} (a (3 B+4 C)) \int \sec ^3(c+d x) \, dx\\ &=\frac {a (3 B+4 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} (a (3 B+4 C)) \int \sec (c+d x) \, dx-\frac {(a (B+C)) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a (3 B+4 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 {a (B+C) \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 77, normalized size = 0.73 \[ \frac {a \left (3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left (8 (B+C) (\cos (2 (c+d x))+2) \sec (c+d x)+6 B \sec ^2(c+d x)+9 B+12 C\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 127, normalized size = 1.20 \[ \frac {3 \, {\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (B + C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, B + 4 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + 6 \, B a\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 188, normalized size = 1.77 \[ \frac {3 \, {\left (3 \, B a + 4 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, B a + 4 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 49 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 28 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 52 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 39 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 171, normalized size = 1.61 \[ \frac {a C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 a B \tan \left (d x +c \right )}{3 d}+\frac {a B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {2 a C \tan \left (d x +c \right )}{3 d}+\frac {a C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 163, normalized size = 1.54 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 3 \, B a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 166, normalized size = 1.57 \[ \frac {\left (-\frac {3\,B\,a}{4}-C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {49\,B\,a}{12}+\frac {7\,C\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {31\,B\,a}{12}-\frac {13\,C\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,B\,a}{4}+3\,C\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,B+4\,C\right )}{4\,d} \]
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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