Optimal. Leaf size=231 \[ -\frac {(A-B-9 C) \sin (c+d x)}{10 d \sqrt {\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(A+B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {(A-B-9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.67, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4221, 3041, 2977, 2978, 2748, 2641, 2639} \[ -\frac {(A-B-9 C) \sin (c+d x)}{10 d \sqrt {\sec (c+d x)} \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(A+B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {(A-B-9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2977
Rule 2978
Rule 3041
Rule 4221
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx\\ &=-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)} \left (\frac {1}{2} a (7 A+3 B-3 C)-\frac {1}{2} a (A-B-9 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x)}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a^2 (4 A+B-6 C)+\frac {1}{2} a^2 (A+4 B+21 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x)}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt {\sec (c+d x)}}-\frac {(A-B-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{4} a^3 (A+B+3 C)+\frac {3}{4} a^3 (A-B-9 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{15 a^6}\\ &=-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x)}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt {\sec (c+d x)}}-\frac {(A-B-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)}}+\frac {\left ((A-B-9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}+\frac {\left ((A+B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}\\ &=\frac {(A-B-9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(A+B+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x)}+\frac {(4 A+B-6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sqrt {\sec (c+d x)}}-\frac {(A-B-9 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.56, size = 188, normalized size = 0.81 \[ \frac {2 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \left (10 (A+B+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 (A-B-9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {1}{8} \left (\sin \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {3}{2} (c+d x)\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) (4 (2 A-7 B-18 C) \cos (c+d x)+3 (A-B-9 C) \cos (2 (c+d x))-7 A-13 B-57 C)\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {\sec \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.95, size = 624, normalized size = 2.70 \[ \frac {\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 )}\, \left (12 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-12 B \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 B \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-108 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-54 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-22 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 B \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+198 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 B \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-114 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-17 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 A +3 B -3 C \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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