Optimal. Leaf size=91 \[ \frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {a^2 x}{4} \]
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Rubi [A] time = 0.13, antiderivative size = 105, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac {a^2 \cos ^3(c+d x)}{6 d}-\frac {\cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{10 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {a^2 x}{4}-\frac {\cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2860
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx &=-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}+\frac {2}{5} \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {\cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{10 d}+\frac {1}{2} a \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a^2 \cos ^3(c+d x)}{6 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {\cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{10 d}+\frac {1}{2} a^2 \int \cos ^2(c+d x) \, dx\\ &=-\frac {a^2 \cos ^3(c+d x)}{6 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{4 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {\cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{10 d}+\frac {1}{4} a^2 \int 1 \, dx\\ &=\frac {a^2 x}{4}-\frac {a^2 \cos ^3(c+d x)}{6 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{4 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {\cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 57, normalized size = 0.63 \[ \frac {a^2 (-90 \cos (c+d x)-25 \cos (3 (c+d x))+3 (-5 \sin (4 (c+d x))+\cos (5 (c+d x))+20 c+20 d x))}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 72, normalized size = 0.79 \[ \frac {12 \, a^{2} \cos \left (d x + c\right )^{5} - 40 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} d x - 15 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 72, normalized size = 0.79 \[ \frac {1}{4} \, a^{2} x + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {3 \, a^{2} \cos \left (d x + c\right )}{8 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 95, normalized size = 1.04 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+2 a^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 69, normalized size = 0.76 \[ -\frac {80 \, a^{2} \cos \left (d x + c\right )^{3} - 16 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.13, size = 262, normalized size = 2.88 \[ \frac {a^2\,x}{4}-\frac {\frac {a^2\,\left (c+d\,x\right )}{4}-3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}-\frac {a^2\,\left (15\,c+15\,d\,x-56\right )}{60}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {5\,a^2\,\left (c+d\,x\right )}{4}-\frac {a^2\,\left (75\,c+75\,d\,x-120\right )}{60}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^2\,\left (c+d\,x\right )}{4}-\frac {a^2\,\left (75\,c+75\,d\,x-160\right )}{60}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {5\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (150\,c+150\,d\,x-80\right )}{60}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {5\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (150\,c+150\,d\,x-480\right )}{60}\right )+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 172, normalized size = 1.89 \[ \begin {cases} \frac {a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} - \frac {2 a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{2} \sin {\relax (c )} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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