Optimal. Leaf size=131 \[ \frac {a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+3 C)+\frac {a C \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac {b (5 A+4 C) \sin (c+d x)}{5 d}+\frac {b C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.20, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3034, 3023, 2748, 2635, 8, 2633} \[ \frac {a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x (4 A+3 C)+\frac {a C \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac {b (5 A+4 C) \sin (c+d x)}{5 d}+\frac {b C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3023
Rule 3034
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^2(c+d x) \left (5 a A+b (5 A+4 C) \cos (c+d x)+5 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^2(c+d x) (5 a (4 A+3 C)+4 b (5 A+4 C) \cos (c+d x)) \, dx\\ &=\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{4} (a (4 A+3 C)) \int \cos ^2(c+d x) \, dx+\frac {1}{5} (b (5 A+4 C)) \int \cos ^3(c+d x) \, dx\\ &=\frac {a (4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{8} (a (4 A+3 C)) \int 1 \, dx-\frac {(b (5 A+4 C)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {1}{8} a (4 A+3 C) x+\frac {b (5 A+4 C) \sin (c+d x)}{5 d}+\frac {a (4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {b C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {b (5 A+4 C) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 89, normalized size = 0.68 \[ \frac {15 a (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))-160 b (A+2 C) \sin ^3(c+d x)+480 b (A+C) \sin (c+d x)+96 b C \sin ^5(c+d x)}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 94, normalized size = 0.72 \[ \frac {15 \, {\left (4 \, A + 3 \, C\right )} a d x + {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, C a \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, A + 3 \, C\right )} a \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, C\right )} b\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 109, normalized size = 0.83 \[ \frac {1}{8} \, {\left (4 \, A a + 3 \, C a\right )} x + \frac {C b \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {C a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A b + 5 \, C b\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (6 \, A b + 5 \, C b\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 117, normalized size = 0.89 \[ \frac {\frac {C b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 113, normalized size = 0.86 \[ \frac {120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 279, normalized size = 2.13 \[ \frac {\left (2\,A\,b-A\,a-\frac {5\,C\,a}{4}+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {16\,A\,b}{3}-2\,A\,a-\frac {C\,a}{2}+\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b}{3}+\frac {116\,C\,b}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (2\,A\,a+\frac {16\,A\,b}{3}+\frac {C\,a}{2}+\frac {8\,C\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A\,a+2\,A\,b+\frac {5\,C\,a}{4}+2\,C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+3\,C\right )}{4\,\left (A\,a+\frac {3\,C\,a}{4}\right )}\right )\,\left (4\,A+3\,C\right )}{4\,d}-\frac {a\,\left (4\,A+3\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.22, size = 279, normalized size = 2.13 \[ \begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C b \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right ) \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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