Optimal. Leaf size=141 \[ \frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {11 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {11 a^2 x}{128} \]
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Rubi [A] time = 0.26, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2873, 2568, 2635, 8, 2565, 14} \[ \frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {11 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {11 a^2 x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^4(c+d x) \sin ^2(c+d x)+2 a^2 \cos ^4(c+d x) \sin ^3(c+d x)+a^2 \cos ^4(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{6} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {11 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{16} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{8} a^2 \int \cos ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {11 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{64} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{16} a^2 \int 1 \, dx\\ &=\frac {a^2 x}{16}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {11 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{128} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac {11 a^2 x}{128}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {11 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 96, normalized size = 0.68 \[ \frac {a^2 (1680 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-560 \sin (6 (c+d x))+105 \sin (8 (c+d x))-10080 \cos (c+d x)-3360 \cos (3 (c+d x))+672 \cos (5 (c+d x))+480 \cos (7 (c+d x))+3360 c+9240 d x)}{107520 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 98, normalized size = 0.70 \[ \frac {3840 \, a^{2} \cos \left (d x + c\right )^{7} - 5376 \, a^{2} \cos \left (d x + c\right )^{5} + 1155 \, a^{2} d x + 35 \, {\left (48 \, a^{2} \cos \left (d x + c\right )^{7} - 136 \, a^{2} \cos \left (d x + c\right )^{5} + 22 \, a^{2} \cos \left (d x + c\right )^{3} + 33 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 140, normalized size = 0.99 \[ \frac {11}{128} \, a^{2} x + \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {3 \, a^{2} \cos \left (d x + c\right )}{32 \, d} + \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 164, normalized size = 1.16 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+2 a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 102, normalized size = 0.72 \[ \frac {6144 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} + 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 105 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{107520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.35, size = 363, normalized size = 2.57 \[ \frac {11\,a^2\,x}{128}-\frac {\frac {11\,a^2\,\left (c+d\,x\right )}{128}-\frac {259\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}-\frac {1103\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {2261\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}-\frac {2261\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {1103\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {259\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {a^2\,\left (1155\,c+1155\,d\,x-3072\right )}{13440}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {11\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (9240\,c+9240\,d\,x-24576\right )}{13440}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {77\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (32340\,c+32340\,d\,x+21504\right )}{13440}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {77\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (32340\,c+32340\,d\,x-107520\right )}{13440}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {385\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (80850\,c+80850\,d\,x-107520\right )}{13440}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {77\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (64680\,c+64680\,d\,x-172032\right )}{13440}\right )+\frac {11\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.02, size = 420, normalized size = 2.98 \[ \begin {cases} \frac {3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {2 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {4 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin ^{2}{\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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