Optimal. Leaf size=187 \[ \frac {8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac {2 a (63 A+57 B+47 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d} \]
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Rubi [A] time = 0.42, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3045, 2968, 3023, 2751, 2647, 2646} \[ \frac {8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac {2 a (63 A+57 B+47 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rule 2968
Rule 3023
Rule 3045
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C)+\frac {3}{2} a (3 B+C) \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 \int (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C) \cos (c+d x)+\frac {3}{2} a (3 B+C) \cos ^2(c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {4 \int (a+a \cos (c+d x))^{3/2} \left (\frac {15}{4} a^2 (3 B+C)+\frac {1}{4} a^2 (63 A-18 B+22 C) \cos (c+d x)\right ) \, dx}{63 a^2}\\ &=\frac {2 (63 A-18 B+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {1}{105} (63 A+57 B+47 C) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {1}{315} (4 a (63 A+57 B+47 C)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 113, normalized size = 0.60 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (756 A+759 B+799 C) \cos (c+d x)+4 (63 A+117 B+137 C) \cos (2 (c+d x))+3276 A+90 B \cos (3 (c+d x))+2964 B+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2689 C)}{1260 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 115, normalized size = 0.61 \[ \frac {2 \, {\left (35 \, C a \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B + 17 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 39 \, B + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + {\left (189 \, A + 156 \, B + 136 \, C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (189 \, A + 156 \, B + 136 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 249, normalized size = 1.33 \[ \frac {1}{2520} \, \sqrt {2} {\left (\frac {35 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {45 \, {\left (2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {126 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {210 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {630 \, {\left (8 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 132, normalized size = 0.71 \[ \frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-180 B -900 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (126 A +504 B +1134 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-315 A -525 B -735 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 A +315 B +315 C \right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 205, normalized size = 1.10 \[ \frac {252 \, {\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 6 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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