Optimal. Leaf size=134 \[ -\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{20 d}+\frac {a^2 (5 A+2 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^2 x (5 A+2 B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d} \]
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Rubi [A] time = 0.17, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{20 d}+\frac {a^2 (5 A+2 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^2 x (5 A+2 B)-\frac {B \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d} \]
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) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}+\frac {1}{5} (5 A+2 B) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{4} (a (5 A+2 B)) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{4} \left (a^2 (5 A+2 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}+\frac {a^2 (5 A+2 B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}+\frac {1}{8} \left (a^2 (5 A+2 B)\right ) \int 1 \, dx\\ &=\frac {1}{8} a^2 (5 A+2 B) x-\frac {a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}+\frac {a^2 (5 A+2 B) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {B \cos ^3(c+d x) (a+a \sin (c+d x))^2}{5 d}-\frac {(5 A+2 B) \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{20 d}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 133, normalized size = 0.99 \[ -\frac {a^2 \cos (c+d x) \left (8 (10 A+7 B) \cos (2 (c+d x))+\frac {60 (5 A+2 B) \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}-135 A \sin (c+d x)+15 A \sin (3 (c+d x))+80 A-30 B \sin (c+d x)+30 B \sin (3 (c+d x))-6 B \cos (4 (c+d x))+62 B\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 95, normalized size = 0.71 \[ \frac {24 \, B a^{2} \cos \left (d x + c\right )^{5} - 80 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{3} + 15 \, {\left (5 \, A + 2 \, B\right )} a^{2} d x - 15 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - {\left (5 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 130, normalized size = 0.97 \[ \frac {B a^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {A a^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {1}{8} \, {\left (5 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + 5 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \cos \left (d x + c\right )}{8 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 182, normalized size = 1.36 \[ \frac {a^{2} A \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )-\frac {2 a^{2} A \left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {B \,a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 134, normalized size = 1.00 \[ -\frac {320 \, A a^{2} \cos \left (d x + c\right )^{3} + 160 \, B a^{2} \cos \left (d x + c\right )^{3} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{2} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 32 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} B a^{2} - 30 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{2}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.47, size = 367, normalized size = 2.74 \[ \frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A+2\,B\right )}{4\,\left (\frac {5\,A\,a^2}{4}+\frac {B\,a^2}{2}\right )}\right )\,\left (5\,A+2\,B\right )}{4\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}-\frac {\frac {4\,A\,a^2}{3}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,A\,a^2}{4}-\frac {B\,a^2}{2}\right )+\frac {14\,B\,a^2}{15}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (4\,A\,a^2+2\,B\,a^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^2}{2}+3\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {7\,A\,a^2}{2}+3\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {3\,A\,a^2}{4}-\frac {B\,a^2}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,A\,a^2+8\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^2}{3}+\frac {8\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {16\,A\,a^2}{3}+\frac {4\,B\,a^2}{3}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.43, size = 371, normalized size = 2.77 \[ \begin {cases} \frac {A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {2 A a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} - \frac {2 B a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {B a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{2} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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