Optimal. Leaf size=163 \[ \frac {a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (4 A+3 C) \sin (c+d x) \cos (c+d x)}{12 d}+\frac {1}{4} a^2 x (4 A+3 C)+\frac {(10 A+3 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{30 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{10 a d} \]
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Rubi [A] time = 0.29, antiderivative size = 163, 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 = {3046, 2968, 3023, 2751, 2644} \[ \frac {a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (4 A+3 C) \sin (c+d x) \cos (c+d x)}{12 d}+\frac {1}{4} a^2 x (4 A+3 C)+\frac {(10 A+3 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{30 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{10 a d} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 2751
Rule 2968
Rule 3023
Rule 3046
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\int \cos (c+d x) (a+a \cos (c+d x))^2 (a (5 A+2 C)+2 a C \cos (c+d x)) \, dx}{5 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^2 \left (a (5 A+2 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx}{5 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{10 a d}+\frac {\int (a+a \cos (c+d x))^2 \left (6 a^2 C+2 a^2 (10 A+3 C) \cos (c+d x)\right ) \, dx}{20 a^2}\\ &=\frac {(10 A+3 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{30 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{10 a d}+\frac {1}{6} (4 A+3 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac {1}{4} a^2 (4 A+3 C) x+\frac {a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (4 A+3 C) \cos (c+d x) \sin (c+d x)}{12 d}+\frac {(10 A+3 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{30 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{10 a d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 97, normalized size = 0.60 \[ \frac {a^2 (30 (14 A+11 C) \sin (c+d x)+120 (A+C) \sin (2 (c+d x))+20 A \sin (3 (c+d x))+240 A d x+45 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+3 C \sin (5 (c+d x))+120 c C+180 C d x)}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 106, normalized size = 0.65 \[ \frac {15 \, {\left (4 \, A + 3 \, C\right )} a^{2} d x + {\left (12 \, C a^{2} \cos \left (d x + c\right )^{4} + 30 \, C a^{2} \cos \left (d x + c\right )^{3} + 4 \, {\left (5 \, A + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right ) + 4 \, {\left (25 \, A + 18 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 129, normalized size = 0.79 \[ \frac {C a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {1}{4} \, {\left (4 \, A a^{2} + 3 \, C a^{2}\right )} x + \frac {{\left (4 \, A a^{2} + 9 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (14 \, A a^{2} + 11 \, C a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 160, normalized size = 0.98 \[ \frac {\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a^{2} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} A \sin \left (d x +c \right )+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 156, normalized size = 0.96 \[ -\frac {80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{2} + 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 240 \, A a^{2} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 277, normalized size = 1.70 \[ \frac {\left (2\,A\,a^2+\frac {3\,C\,a^2}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {28\,A\,a^2}{3}+7\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {56\,A\,a^2}{3}+\frac {72\,C\,a^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {52\,A\,a^2}{3}+9\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (6\,A\,a^2+\frac {13\,C\,a^2}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^2\,\left (4\,A+3\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,A+3\,C\right )}{2\,\left (2\,A\,a^2+\frac {3\,C\,a^2}{2}\right )}\right )\,\left (4\,A+3\,C\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 350, normalized size = 2.15 \[ \begin {cases} A a^{2} x \sin ^{2}{\left (c + d x \right )} + A a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {8 C a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {2 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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