Optimal. Leaf size=67 \[ \frac {(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {4 (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {(a+a \sin (c+d x))^8}{8 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 58, normalized size = 0.87 \[ -\frac {a^3 (\sin (c+d x)+1)^3 \left (21 \sin ^2(c+d x)-54 \sin (c+d x)+37\right ) \cos ^6(c+d x)}{168 d (\sin (c+d x)-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 85, normalized size = 1.27 \[ \frac {21 \, a^{3} \cos \left (d x + c\right )^{8} - 112 \, a^{3} \cos \left (d x + c\right )^{6} - 8 \, {\left (9 \, a^{3} \cos \left (d x + c\right )^{6} - 6 \, a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 134, normalized size = 2.00 \[ \frac {a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {5 \, a^{3} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {25 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {33 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {3 \, a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a^{3} \sin \left (5 \, d x + 5 \, c\right )}{64 \, d} + \frac {17 \, a^{3} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {55 \, a^{3} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 133, normalized size = 1.99 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) a^{3}}{2}+\frac {a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 108, normalized size = 1.61 \[ \frac {21 \, a^{3} \sin \left (d x + c\right )^{8} + 72 \, a^{3} \sin \left (d x + c\right )^{7} + 28 \, a^{3} \sin \left (d x + c\right )^{6} - 168 \, a^{3} \sin \left (d x + c\right )^{5} - 210 \, a^{3} \sin \left (d x + c\right )^{4} + 56 \, a^{3} \sin \left (d x + c\right )^{3} + 252 \, a^{3} \sin \left (d x + c\right )^{2} + 168 \, a^{3} \sin \left (d x + c\right )}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 106, normalized size = 1.58 \[ \frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{6}-a^3\,{\sin \left (c+d\,x\right )}^5-\frac {5\,a^3\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a^3\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^2}{2}+a^3\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.40, size = 196, normalized size = 2.93 \[ \begin {cases} \frac {8 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {4 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{3} \cos ^{8}{\left (c + d x \right )}}{24 d} - \frac {a^{3} \cos ^{6}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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