Optimal. Leaf size=236 \[ \frac {4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt {\sec (c+d x)}}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac {5}{2}}(c+d x)}-\frac {5 (A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac {4 (5 A+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {(A+C) \sin (c+d x)}{3 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.46, antiderivative size = 236, 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, 2635, 2641, 2639} \[ \frac {4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt {\sec (c+d x)}}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac {5}{2}}(c+d x)}-\frac {5 (A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac {4 (5 A+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {(A+C) \sin (c+d x)}{3 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^2} \]
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))^2 \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))^2} \, dx\\ &=-\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \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 (A+7 C)+\frac {1}{2} a (5 A+11 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (-\frac {15}{2} a^2 (A+3 C)+2 a^2 (5 A+14 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}-\frac {\left (5 (A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{2 a^2}+\frac {\left (2 (5 A+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{3 a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt {\sec (c+d x)}}-\frac {\left (5 (A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a^2}+\frac {\left (2 (5 A+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^2}\\ &=\frac {4 (5 A+14 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a^2 d}-\frac {5 (A+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 a^2 d}-\frac {(A+C) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(A+3 C) \sin (c+d x)}{a^2 d (1+\cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.94, size = 813, normalized size = 3.44 \[ -\frac {4 \sqrt {2} A 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)}} \csc \left (\frac {c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _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 ) \sec \left (\frac {c}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2}-\frac {56 \sqrt {2} 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)}} \csc \left (\frac {c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _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 ) \sec \left (\frac {c}{2}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d (\cos (c+d x) a+a)^2}-\frac {10 A \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sqrt {\sec (c+d x)} \sin (c) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2}-\frac {10 C \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sqrt {\sec (c+d x)} \sin (c) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{d (\cos (c+d x) a+a)^2}+\frac {\sqrt {\sec (c+d x)} \left (-\frac {2 \sec \left (\frac {c}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}-\frac {2 (A+C) \tan \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}+\frac {4 \sec \left (\frac {c}{2}\right ) \left (7 A \sin \left (\frac {d x}{2}\right )+13 C \sin \left (\frac {d x}{2}\right )\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}-\frac {(20 \cos (2 c) A+60 A+151 C+73 C \cos (2 c)) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{10 d}-\frac {8 C \cos (2 d x) \sin (2 c)}{3 d}+\frac {2 C \cos (3 d x) \sin (3 c)}{5 d}+\frac {2 (20 A+73 C) \cos (c) \sin (d x)}{5 d}-\frac {8 C \cos (2 c) \sin (2 d x)}{3 d}+\frac {2 C \cos (3 c) \sin (3 d x)}{5 d}+\frac {4 (7 A+13 C) \tan \left (\frac {c}{2}\right )}{3 d}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{(\cos (c+d x) a+a)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.73, size = 451, normalized size = 1.91 \[ \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 (-96 C \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+352 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 A \left (\cos ^{3}\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 )+120 A \left (\cos ^{3}\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 )-120 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+150 C \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 ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 C \left (\cos ^{3}\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 )-190 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-266 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+135 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 A -5 C \right )}{30 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
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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