Optimal. Leaf size=343 \[ \frac {4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^3 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 (13 B+6 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 0.85, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4221, 3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639} \[ \frac {4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (143 A+195 B+145 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^3 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 (13 B+6 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3045
Rule 4221
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+5 C)+\frac {1}{2} a (13 B+6 C) \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+65 B+85 C)+\frac {1}{4} a^2 (143 A+195 B+145 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac {1}{4} a^3 (1001 A+780 B+745 C)+\frac {5}{4} a^3 (286 A+273 B+236 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{4} a^4 (1001 A+780 B+745 C)+\left (\frac {5}{4} a^4 (286 A+273 B+236 C)+\frac {1}{4} a^4 (1001 A+780 B+745 C)\right ) \cos (c+d x)+\frac {5}{4} a^4 (286 A+273 B+236 C) \cos ^2(c+d x)\right ) \, dx}{1287 a}\\ &=\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {117}{8} a^4 (121 A+105 B+95 C)+\frac {77}{8} a^4 (221 A+195 B+175 C) \cos (c+d x)\right ) \, dx}{9009 a}\\ &=\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{77} \left (2 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{117} \left (2 a^3 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (2 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{195} \left (2 a^3 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^3 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+195 B+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.16, size = 197, normalized size = 0.57 \[ \frac {a^3 \sqrt {\sec (c+d x)} \left (2 \sin (2 (c+d x)) (154 (3926 A+4290 B+4525 C) \cos (c+d x)+5 (936 (33 A+49 B+59 C) \cos (2 (c+d x))+77 (52 A+156 B+245 C) \cos (3 (c+d x))+3 (60632 A+546 (B+3 C) \cos (4 (c+d x))+58422 B+231 C \cos (5 (c+d x))+56290 C)))+24960 (121 A+105 B+95 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+29568 (221 A+195 B+175 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{1441440 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C a^{3} \cos \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.80, size = 576, normalized size = 1.68 \[ -\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (-221760 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (131040 B +1058400 C \right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-80080 A -567840 B -2122400 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (314600 A +1004640 B +2331040 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-487916 A -939120 B -1535860 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (386386 A +510510 B +633710 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-105534 A -114660 B -121230 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23595 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20475 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-45045 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+18525 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int \frac {A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \cos {\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \cos {\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B \cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \cos ^{4}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \cos ^{4}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \cos ^{5}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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