Optimal. Leaf size=109 \[ \frac {(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac {3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac {13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac {4 (a \sin (c+d x)+a)^5}{5 a^3 d} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac {(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac {3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac {13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac {2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac {4 (a \sin (c+d x)+a)^5}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^2 x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^4 (a+x)^4-12 a^3 (a+x)^5+13 a^2 (a+x)^6-6 a (a+x)^7+(a+x)^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac {2 (a+a \sin (c+d x))^6}{a^4 d}+\frac {13 (a+a \sin (c+d x))^7}{7 a^5 d}-\frac {3 (a+a \sin (c+d x))^8}{4 a^6 d}+\frac {(a+a \sin (c+d x))^9}{9 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 99, normalized size = 0.91 \[ -\frac {a^2 (-16380 \sin (c+d x)+1680 \sin (3 (c+d x))+2016 \sin (5 (c+d x))+270 \sin (7 (c+d x))-70 \sin (9 (c+d x))+7560 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-840 \cos (6 (c+d x))-315 \cos (8 (c+d x)))}{161280 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 98, normalized size = 0.90 \[ \frac {315 \, a^{2} \cos \left (d x + c\right )^{8} - 420 \, a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (35 \, a^{2} \cos \left (d x + c\right )^{8} - 95 \, a^{2} \cos \left (d x + c\right )^{6} + 12 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{2} + 32 \, a^{2}\right )} \sin \left (d x + c\right )}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 151, normalized size = 1.39 \[ \frac {a^{2} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac {a^{2} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a^{2} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac {3 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {a^{2} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {3 \, a^{2} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a^{2} \sin \left (3 \, d x + 3 \, c\right )}{96 \, d} + \frac {13 \, a^{2} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 156, normalized size = 1.43 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+2 a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 97, normalized size = 0.89 \[ \frac {140 \, a^{2} \sin \left (d x + c\right )^{9} + 315 \, a^{2} \sin \left (d x + c\right )^{8} - 180 \, a^{2} \sin \left (d x + c\right )^{7} - 840 \, a^{2} \sin \left (d x + c\right )^{6} - 252 \, a^{2} \sin \left (d x + c\right )^{5} + 630 \, a^{2} \sin \left (d x + c\right )^{4} + 420 \, a^{2} \sin \left (d x + c\right )^{3}}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.76, size = 96, normalized size = 0.88 \[ \frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a^2\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {a^2\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {2\,a^2\,{\sin \left (c+d\,x\right )}^6}{3}-\frac {a^2\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a^2\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {a^2\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.76, size = 190, normalized size = 1.74 \[ \begin {cases} \frac {8 a^{2} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \cos ^{8}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{2}{\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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