Optimal. Leaf size=165 \[ -\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{42 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (7 A+2 B)-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.19, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{42 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (7 A+2 B)-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2860
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {1}{7} (7 A+2 B) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{6} (a (7 A+2 B)) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{6} \left (a^2 (7 A+2 B)\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{8} \left (a^2 (7 A+2 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{16} \left (a^2 (7 A+2 B)\right ) \int 1 \, dx\\ &=\frac {1}{16} a^2 (7 A+2 B) x-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}\\ \end {align*}
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Mathematica [A] time = 1.73, size = 171, normalized size = 1.04 \[ -\frac {a^2 \cos (c+d x) \left ((672 A+447 B) \cos (2 (c+d x))+6 (28 A+13 B) \cos (4 (c+d x))+\frac {420 (7 A+2 B) \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}-1645 A \sin (c+d x)-140 A \sin (3 (c+d x))+35 A \sin (5 (c+d x))+504 A-350 B \sin (c+d x)+140 B \sin (3 (c+d x))+70 B \sin (5 (c+d x))-15 B \cos (6 (c+d x))+354 B\right )}{3360 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 115, normalized size = 0.70 \[ \frac {240 \, B a^{2} \cos \left (d x + c\right )^{7} - 672 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{5} + 105 \, {\left (7 \, A + 2 \, B\right )} a^{2} d x - 35 \, {\left (8 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (7 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - 3 \, {\left (7 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 192, normalized size = 1.16 \[ \frac {B a^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (7 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + 3 \, B a^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (8 \, A a^{2} + 5 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (16 \, A a^{2} + 11 \, B a^{2}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (A a^{2} - 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (17 \, A a^{2} + 2 \, B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 215, normalized size = 1.30 \[ \frac {a^{2} A \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-\frac {2 a^{2} A \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {B \,a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 171, normalized size = 1.04 \[ -\frac {2688 \, A a^{2} \cos \left (d x + c\right )^{5} + 1344 \, B a^{2} \cos \left (d x + c\right )^{5} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 192 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} B a^{2} - 70 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{2}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.00, size = 494, normalized size = 2.99 \[ \frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+2\,B\right )}{8\,\left (\frac {7\,A\,a^2}{8}+\frac {B\,a^2}{4}\right )}\right )\,\left (7\,A+2\,B\right )}{8\,d}-\frac {a^2\,\left (7\,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 )}{8\,d}-\frac {\frac {4\,A\,a^2}{5}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,A\,a^2}{8}-\frac {B\,a^2}{4}\right )+\frac {18\,B\,a^2}{35}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (4\,A\,a^2+2\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (12\,A\,a^2+2\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\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}{5}+\frac {8\,B\,a^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {9\,A\,a^2}{8}-\frac {B\,a^2}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (16\,A\,a^2+16\,B\,a^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {29\,A\,a^2}{6}+\frac {11\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {29\,A\,a^2}{6}+\frac {11\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {44\,A\,a^2}{5}+\frac {14\,B\,a^2}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {23\,A\,a^2}{24}-\frac {31\,B\,a^2}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {23\,A\,a^2}{24}-\frac {31\,B\,a^2}{12}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\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 = 6.70, size = 539, normalized size = 3.27 \[ \begin {cases} \frac {A a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 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 ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 A a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {B a^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {B a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {2 B a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {B a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{2} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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