Optimal. Leaf size=117 \[ \frac {(8 A+7 C) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 (8 A+7 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} x (8 A+7 C)+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d} \]
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Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ \frac {(8 A+7 C) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 (8 A+7 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} x (8 A+7 C)+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3014
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} (8 A+7 C) \int \cos ^6(c+d x) \, dx\\ &=\frac {(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} (5 (8 A+7 C)) \int \cos ^4(c+d x) \, dx\\ &=\frac {5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} (5 (8 A+7 C)) \int \cos ^2(c+d x) \, dx\\ &=\frac {5 (8 A+7 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} (5 (8 A+7 C)) \int 1 \, dx\\ &=\frac {5}{128} (8 A+7 C) x+\frac {5 (8 A+7 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 93, normalized size = 0.79 \[ \frac {48 (15 A+14 C) \sin (2 (c+d x))+24 (6 A+7 C) \sin (4 (c+d x))+16 A \sin (6 (c+d x))+960 A c+960 A d x+32 C \sin (6 (c+d x))+3 C \sin (8 (c+d x))+840 c C+840 C d x}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 85, normalized size = 0.73 \[ \frac {15 \, {\left (8 \, A + 7 \, C\right )} d x + {\left (48 \, C \cos \left (d x + c\right )^{7} + 8 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 87, normalized size = 0.74 \[ \frac {5}{128} \, {\left (8 \, A + 7 \, C\right )} x + \frac {C \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (A + 2 \, C\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, A + 7 \, C\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, A + 14 \, C\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 106, normalized size = 0.91 \[ \frac {C \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 130, normalized size = 1.11 \[ \frac {15 \, {\left (d x + c\right )} {\left (8 \, A + 7 \, C\right )} + \frac {15 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{7} + 55 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{5} + 73 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (88 \, A + 93 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 119, normalized size = 1.02 \[ x\,\left (\frac {5\,A}{16}+\frac {35\,C}{128}\right )+\frac {\left (\frac {5\,A}{16}+\frac {35\,C}{128}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^7+\left (\frac {55\,A}{48}+\frac {385\,C}{384}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {73\,A}{48}+\frac {511\,C}{384}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {11\,A}{16}+\frac {93\,C}{128}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8+4\,{\mathrm {tan}\left (c+d\,x\right )}^6+6\,{\mathrm {tan}\left (c+d\,x\right )}^4+4\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.53, size = 354, normalized size = 3.03 \[ \begin {cases} \frac {5 A x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 A x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 A x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 A x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 A \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 A \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 A \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {35 C x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 C x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 C x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 C x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 C x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 C \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 C \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 C \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 C \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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