Optimal. Leaf size=249 \[ \frac {(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {(9 A+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac {2 C \sin (c+d x)}{3 a d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.60, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4221, 3042, 2977, 2748, 2639, 2635, 2641} \[ \frac {(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac {(A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {(9 A+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac {2 C \sin (c+d x)}{3 a d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2977
Rule 3042
Rule 4221
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx\\ &=-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {1}{2} a (3 A-7 C)+\frac {1}{2} a (3 A+13 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 C \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-25 a^2 C+\frac {3}{2} a^2 (3 A+23 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 C \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (-\frac {3}{4} a^3 (9 A+119 C)+\frac {45}{4} a^3 (A+11 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 C \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (3 (A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{4 a^3}-\frac {\left ((9 A+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac {(9 A+119 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+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 C \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x)}+\frac {(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}+\frac {\left ((A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac {(9 A+119 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+11 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{2 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 C \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}-\frac {(9 A+119 C) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x)}+\frac {(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 4.44, size = 573, normalized size = 2.30 \[ \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left ((156 A+1961 C) \cos \left (\frac {1}{2} (c-d x)\right )+(114 A+1609 C) \cos \left (\frac {1}{2} (3 c+d x)\right )+90 A \cos \left (\frac {1}{2} (c+3 d x)\right )+45 A \cos \left (\frac {1}{2} (5 c+3 d x)\right )+27 A \cos \left (\frac {1}{2} (3 c+5 d x)\right )+1165 C \cos \left (\frac {1}{2} (c+3 d x)\right )+620 C \cos \left (\frac {1}{2} (5 c+3 d x)\right )+292 C \cos \left (\frac {1}{2} (3 c+5 d x)\right )+65 C \cos \left (\frac {1}{2} (7 c+5 d x)\right )+5 C \cos \left (\frac {1}{2} (5 c+7 d x)\right )-5 C \cos \left (\frac {1}{2} (9 c+7 d x)\right )\right )}{8 \sqrt {\sec (c+d x)}}+18 \sqrt {2} A \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right )+60 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+238 \sqrt {2} C \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right )+660 C \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.22, size = 465, normalized size = 1.87 \[ -\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 (160 C \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+108 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+30 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 )+54 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 )+468 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+330 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 )+714 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 )-198 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1058 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+114 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+474 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-47 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A +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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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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