Optimal. Leaf size=178 \[ \frac {\left (5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (2 a^2 C+b^2 (5 A+4 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac {a b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x (4 A+3 C)+\frac {a b C \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.30, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3050, 3033, 3023, 2734} \[ \frac {\left (5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (2 a^2 C+b^2 (5 A+4 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac {a b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x (4 A+3 C)+\frac {a b C \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rule 3033
Rule 3050
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos (c+d x) (a+b \cos (c+d x)) \left (a (5 A+2 C)+b (5 A+4 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a b C \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos (c+d x) \left (4 a^2 (5 A+2 C)+10 a b (4 A+3 C) \cos (c+d x)+4 \left (2 a^2 C+b^2 (5 A+4 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left (2 a^2 C+b^2 (5 A+4 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {a b C \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{60} \int \cos (c+d x) \left (4 \left (5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right )+30 a b (4 A+3 C) \cos (c+d x)\right ) \, dx\\ &=\frac {1}{4} a b (4 A+3 C) x+\frac {\left (5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {a b (4 A+3 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {\left (2 a^2 C+b^2 (5 A+4 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {a b C \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 126, normalized size = 0.71 \[ \frac {30 \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x)+5 \left (4 a^2 C+4 A b^2+5 b^2 C\right ) \sin (3 (c+d x))+60 a b (4 A+3 C) (c+d x)+120 a b (A+C) \sin (2 (c+d x))+15 a b C \sin (4 (c+d x))+3 b^2 C \sin (5 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 123, normalized size = 0.69 \[ \frac {15 \, {\left (4 \, A + 3 \, C\right )} a b d x + {\left (12 \, C b^{2} \cos \left (d x + c\right )^{4} + 30 \, C a b \cos \left (d x + c\right )^{3} + 15 \, {\left (4 \, A + 3 \, C\right )} a b \cos \left (d x + c\right ) + 20 \, {\left (3 \, A + 2 \, C\right )} a^{2} + 8 \, {\left (5 \, A + 4 \, C\right )} b^{2} + 4 \, {\left (5 \, C a^{2} + {\left (5 \, A + 4 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 142, normalized size = 0.80 \[ \frac {C b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {C a b \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {1}{4} \, {\left (4 \, A a b + 3 \, C a b\right )} x + \frac {{\left (4 \, C a^{2} + 4 \, A b^{2} + 5 \, C b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a b + C a b\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (8 \, A a^{2} + 6 \, C a^{2} + 6 \, A b^{2} + 5 \, C b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 158, normalized size = 0.89 \[ \frac {a^{2} A \sin \left (d x +c \right )+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 A a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C a b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {b^{2} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 154, normalized size = 0.87 \[ -\frac {80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b + 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{2} - 16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{2} - 240 \, A a^{2} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 371, normalized size = 2.08 \[ \frac {\left (2\,A\,a^2+2\,A\,b^2+2\,C\,a^2+2\,C\,b^2-2\,A\,a\,b-\frac {5\,C\,a\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (8\,A\,a^2+\frac {16\,A\,b^2}{3}+\frac {16\,C\,a^2}{3}+\frac {8\,C\,b^2}{3}-4\,A\,a\,b-C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,A\,a^2+\frac {20\,A\,b^2}{3}+\frac {20\,C\,a^2}{3}+\frac {116\,C\,b^2}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (8\,A\,a^2+\frac {16\,A\,b^2}{3}+\frac {16\,C\,a^2}{3}+\frac {8\,C\,b^2}{3}+4\,A\,a\,b+C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^2+2\,A\,b^2+2\,C\,a^2+2\,C\,b^2+2\,A\,a\,b+\frac {5\,C\,a\,b}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+3\,C\right )}{2\,\left (2\,A\,a\,b+\frac {3\,C\,a\,b}{2}\right )}\right )\,\left (4\,A+3\,C\right )}{2\,d}-\frac {a\,b\,\left (4\,A+3\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.39, size = 350, normalized size = 1.97 \[ \begin {cases} \frac {A a^{2} \sin {\left (c + d x \right )}}{d} + A a b x \sin ^{2}{\left (c + d x \right )} + A a b x \cos ^{2}{\left (c + d x \right )} + \frac {A a b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {2 A b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a b x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 C a b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a b x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {3 C a b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {5 C a b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {8 C b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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