Optimal. Leaf size=257 \[ -\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{35 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 (7 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^3}{d} \]
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Rubi [A] time = 0.68, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4221, 3044, 2976, 2968, 3023, 2748, 2641, 2639} \[ -\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{35 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^3}{d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3044
Rule 4221
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (A+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 \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^3 \left (3 a A-\frac {1}{2} a (7 A-C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{a}\\ &=-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\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 (35 A+C)-\frac {1}{4} a^2 (35 A-11 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{7 a}\\ &=-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x)) \left (\frac {1}{2} a^3 (35 A+4 C)-\frac {1}{4} a^3 (35 A-41 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{35 a}\\ &=-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a^4 (35 A+4 C)+\left (-\frac {1}{4} a^4 (35 A-41 C)+\frac {1}{2} a^4 (35 A+4 C)\right ) \cos (c+d x)-\frac {1}{4} a^4 (35 A-41 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{35 a}\\ &=-\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a^4 (35 A+13 C)+\frac {21}{8} a^4 (5 A+7 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{105 a}\\ &=-\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {1}{5} \left (2 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (35 A+13 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 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 1.93, size = 218, normalized size = 0.85 \[ \frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (5 A+7 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+80 (35 A+13 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+840 A \sin (c+d x)+140 A \sin (2 (c+d x))+1680 i A \cos (c+d x)+126 C \sin (c+d x)+550 C \sin (2 (c+d x))+126 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+2352 i C \cos (c+d x)\right )}{420 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.14, size = 569, normalized size = 2.21 \[ -\frac {4 a^{3} \left (120 C \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 )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-432 C \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 )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14 \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 )}\, \left (5 A +43 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 )-4 \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 )}\, \left (35 A +52 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 )+175 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 ) \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 )}-105 A \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 )}\, \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 )+65 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}\, \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 )}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 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}\, \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 )}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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