Optimal. Leaf size=271 \[ \frac {4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (3 B+2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.78, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4221, 3045, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac {4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (3 B+2 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3045
Rule 4221
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^3 \left (\frac {1}{2} a (9 A+C)+\frac {3}{2} a (3 B+2 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{9 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^2 (63 A+9 B+13 C)+\frac {1}{4} a^2 (63 A+99 B+73 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{63 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x)) \left (\frac {3}{4} a^3 (63 A+24 B+23 C)+\frac {9}{4} a^3 (42 A+41 B+32 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{315 a}\\ &=\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^4 (63 A+24 B+23 C)+\left (\frac {3}{4} a^4 (63 A+24 B+23 C)+\frac {9}{4} a^4 (42 A+41 B+32 C)\right ) \cos (c+d x)+\frac {9}{4} a^4 (42 A+41 B+32 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{315 a}\\ &=\frac {4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {45}{8} a^4 (21 A+13 B+11 C)+\frac {63}{8} a^4 (27 A+21 B+17 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{945 a}\\ &=\frac {4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {1}{21} \left (2 a^3 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (2 a^3 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^3 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 (3 B+2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {2 (63 A+99 B+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.57, size = 153, normalized size = 0.56 \[ \frac {a^3 \sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) (7 (36 A+108 B+151 C) \cos (c+d x)+5 (252 A+18 (B+3 C) \cos (2 (c+d x))+330 B+7 C \cos (3 (c+d x))+318 C))+240 (21 A+13 B+11 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+336 (27 A+21 B+17 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.71, size = 514, normalized size = 1.90 \[ -\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 (-560 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (360 B +2200 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 (-252 A -1296 B -3412 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 (882 A +1806 B +2702 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 (-378 A -624 B -738 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 )+315 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 )-567 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 )+195 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 )-441 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 )+165 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 )-357 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 )}{315 \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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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