Optimal. Leaf size=150 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 b x}{128} \]
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Rubi [A] time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2748, 2633, 2635, 8} \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 b x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx &=a \int \cos ^7(c+d x) \, dx+b \int \cos ^8(c+d x) \, dx\\ &=\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} (7 b) \int \cos ^6(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {a \sin (c+d x)}{d}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}+\frac {1}{48} (35 b) \int \cos ^4(c+d x) \, dx\\ &=\frac {a \sin (c+d x)}{d}+\frac {35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}+\frac {1}{64} (35 b) \int \cos ^2(c+d x) \, dx\\ &=\frac {a \sin (c+d x)}{d}+\frac {35 b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}+\frac {1}{128} (35 b) \int 1 \, dx\\ &=\frac {35 b x}{128}+\frac {a \sin (c+d x)}{d}+\frac {35 b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 135, normalized size = 0.90 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {35 b (c+d x)}{128 d}+\frac {7 b \sin (2 (c+d x))}{32 d}+\frac {7 b \sin (4 (c+d x))}{128 d}+\frac {b \sin (6 (c+d x))}{96 d}+\frac {b \sin (8 (c+d x))}{1024 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 97, normalized size = 0.65 \[ \frac {3675 \, b d x + {\left (1680 \, b \cos \left (d x + c\right )^{7} + 1920 \, a \cos \left (d x + c\right )^{6} + 1960 \, b \cos \left (d x + c\right )^{5} + 2304 \, a \cos \left (d x + c\right )^{4} + 2450 \, b \cos \left (d x + c\right )^{3} + 3072 \, a \cos \left (d x + c\right )^{2} + 3675 \, b \cos \left (d x + c\right ) + 6144 \, a\right )} \sin \left (d x + c\right )}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 122, normalized size = 0.81 \[ \frac {35}{128} \, b x + \frac {b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {7 \, b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {35 \, a \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 0.67 \[ \frac {b \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 105, normalized size = 0.70 \[ -\frac {3072 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a + 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{107520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.23, size = 175, normalized size = 1.17 \[ \frac {35\,b\,x}{128}+\frac {\left (2\,a-\frac {93\,b}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (6\,a-\frac {91\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {106\,a}{5}-\frac {1799\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1026\,a}{35}+\frac {1085\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1026\,a}{35}-\frac {1085\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {106\,a}{5}+\frac {1799\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (6\,a+\frac {91\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a+\frac {93\,b}{64}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.96, size = 286, normalized size = 1.91 \[ \begin {cases} \frac {16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {35 b x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 b x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 b \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 b \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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