Optimal. Leaf size=225 \[ -\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac {a \left (-6 a^2 C+100 A b^2+71 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac {1}{8} a x \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right )-\frac {\left (3 a^4 C-4 a^2 b^2 (20 A+13 C)-4 b^4 (5 A+4 C)\right ) \sin (c+d x)}{30 b d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d} \]
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Rubi [A] time = 0.34, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3024, 2753, 2734} \[ -\frac {\left (-4 a^2 b^2 (20 A+13 C)+3 a^4 C-4 b^4 (5 A+4 C)\right ) \sin (c+d x)}{30 b d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac {a \left (-6 a^2 C+100 A b^2+71 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac {1}{8} a x \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3024
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac {\int (a+b \cos (c+d x))^3 (b (5 A+4 C)-a C \cos (c+d x)) \, dx}{5 b}\\ &=-\frac {a C (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac {C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac {\int (a+b \cos (c+d x))^2 \left (a b (20 A+13 C)-\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) \cos (c+d x)\right ) \, dx}{20 b}\\ &=-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 b d}-\frac {a C (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac {C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}+\frac {\int (a+b \cos (c+d x)) \left (b \left (8 b^2 (5 A+4 C)+a^2 (60 A+33 C)\right )+a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \cos (c+d x)\right ) \, dx}{60 b}\\ &=\frac {1}{8} a \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right ) x-\frac {\left (3 a^4 C-4 b^4 (5 A+4 C)-4 a^2 b^2 (20 A+13 C)\right ) \sin (c+d x)}{30 b d}+\frac {a \left (100 A b^2-6 a^2 C+71 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{120 d}-\frac {\left (3 a^2 C-4 b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 b d}-\frac {a C (a+b \cos (c+d x))^3 \sin (c+d x)}{20 b d}+\frac {C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 160, normalized size = 0.71 \[ \frac {60 a (c+d x) \left (4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right )+60 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sin (c+d x)+10 b \left (12 a^2 C+4 A b^2+5 b^2 C\right ) \sin (3 (c+d x))+120 a \left (C \left (a^2+3 b^2\right )+3 A b^2\right ) \sin (2 (c+d x))+45 a b^2 C \sin (4 (c+d x))+6 b^3 C \sin (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 153, normalized size = 0.68 \[ \frac {15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2}\right )} d x + {\left (24 \, C b^{3} \cos \left (d x + c\right )^{4} + 90 \, C a b^{2} \cos \left (d x + c\right )^{3} + 120 \, {\left (3 \, A + 2 \, C\right )} a^{2} b + 16 \, {\left (5 \, A + 4 \, C\right )} b^{3} + 8 \, {\left (15 \, C a^{2} b + {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, C a^{3} + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 174, normalized size = 0.77 \[ \frac {C b^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {3 \, C a b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (8 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 9 \, C a b^{2}\right )} x + \frac {{\left (12 \, C a^{2} b + 4 \, A b^{3} + 5 \, C b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (C a^{3} + 3 \, A a b^{2} + 3 \, C a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (24 \, A a^{2} b + 18 \, C a^{2} b + 6 \, A b^{3} + 5 \, C b^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 201, normalized size = 0.89 \[ \frac {\frac {b^{3} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 C a \,b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{2} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 A a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \,a^{2} b \sin \left (d x +c \right )+A \,a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 194, normalized size = 0.86 \[ \frac {480 \, {\left (d x + c\right )} A a^{3} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} + 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{3} + 1440 \, A a^{2} b \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.86, size = 488, normalized size = 2.17 \[ \frac {\left (2\,A\,b^3-C\,a^3+2\,C\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b-\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {16\,A\,b^3}{3}-2\,C\,a^3+\frac {8\,C\,b^3}{3}-6\,A\,a\,b^2+24\,A\,a^2\,b-\frac {3\,C\,a\,b^2}{2}+16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b^3}{3}+\frac {116\,C\,b^3}{15}+36\,A\,a^2\,b+20\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {16\,A\,b^3}{3}+2\,C\,a^3+\frac {8\,C\,b^3}{3}+6\,A\,a\,b^2+24\,A\,a^2\,b+\frac {3\,C\,a\,b^2}{2}+16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,b^3+C\,a^3+2\,C\,b^3+3\,A\,a\,b^2+6\,A\,a^2\,b+\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+12\,A\,b^2+4\,C\,a^2+9\,C\,b^2\right )}{4\,d}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+12\,A\,b^2+4\,C\,a^2+9\,C\,b^2\right )}{4\,\left (2\,A\,a^3+C\,a^3+3\,A\,a\,b^2+\frac {9\,C\,a\,b^2}{4}\right )}\right )\,\left (8\,A\,a^2+12\,A\,b^2+4\,C\,a^2+9\,C\,b^2\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.68, size = 440, normalized size = 1.96 \[ \begin {cases} A a^{3} x + \frac {3 A a^{2} b \sin {\left (c + d x \right )}}{d} + \frac {3 A a b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C a^{2} b \sin ^{3}{\left (c + d x \right )}}{d} + \frac {3 C a^{2} b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 C a b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 C a b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C b^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C b^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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