Optimal. Leaf size=87 \[ -\frac {(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac {8 (a \sin (c+d x)+a)^5}{5 a^4 d} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac {(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac {8 (a \sin (c+d x)+a)^5}{5 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^4-12 a^2 (a+x)^5+6 a (a+x)^6-(a+x)^7\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {8 (a+a \sin (c+d x))^5}{5 a^4 d}-\frac {2 (a+a \sin (c+d x))^6}{a^5 d}+\frac {6 (a+a \sin (c+d x))^7}{7 a^6 d}-\frac {(a+a \sin (c+d x))^8}{8 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.85 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}-\frac {a \cos ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 62, normalized size = 0.71 \[ -\frac {35 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.69, size = 118, normalized size = 1.36 \[ -\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {35 \, a \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 56, normalized size = 0.64 \[ \frac {-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{8}+\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 92, normalized size = 1.06 \[ -\frac {35 \, a \sin \left (d x + c\right )^{8} + 40 \, a \sin \left (d x + c\right )^{7} - 140 \, a \sin \left (d x + c\right )^{6} - 168 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3} - 140 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right )}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 90, normalized size = 1.03 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}-a\,{\sin \left (c+d\,x\right )}^3+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.29, size = 105, normalized size = 1.21 \[ \begin {cases} \frac {16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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