Optimal. Leaf size=590 \[ \frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{315 a d}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{315 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-3 a^3 (49 A-25 B+63 C)+3 a^2 b (13 A-57 B+21 C)+6 a b^2 (A-3 B)+8 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (3 a B+A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{21 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d} \]
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Rubi [A] time = 2.22, antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4221, 3047, 3055, 2998, 2816, 2994} \[ \frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{315 a d}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{315 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (3 a B+A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{21 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2816
Rule 2994
Rule 2998
Rule 3047
Rule 3055
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {3}{2} (A b+3 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac {1}{2} b (4 A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 (A b+3 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{63} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} \left (92 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \cos (c+d x)+\frac {1}{4} b (40 A b+36 a B+63 b C) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 (A b+3 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{8} \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right )+\frac {1}{8} a \left (396 a b B+21 a^2 (7 A+9 C)+b^2 (209 A+315 C)\right ) \cos (c+d x)+\frac {1}{4} b \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a}\\ &=-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 (A b+3 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{16} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right )+\frac {3}{16} a \left (2 A b^3+75 a^3 B+153 a b^2 B+6 a^2 b (31 A+42 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{945 a^2}\\ &=-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 (A b+3 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a^2}+\frac {\left ((a-b) \left (8 A b^3+6 a b^2 (A-3 B)+3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{315 a^2}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (8 A b^3+6 a b^2 (A-3 B)+3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac {2 (A b+3 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 20.65, size = 801, normalized size = 1.36 \[ \frac {2 \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (-\left ((a+b) \left (21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+8 A b^4\right ) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )\right )+a (a+b) \left (3 (49 A+25 B+63 C) a^3+3 b (13 A+57 B+21 C) a^2-6 b^2 (A+3 B) a+8 A b^3\right ) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )+\left (21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+8 A b^4\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )-1\right ) \left (a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b\right )\right )}{315 a^3 d \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}}}+\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{63} (10 A b \sin (c+d x)+9 a B \sin (c+d x)) \sec ^3(c+d x)+\frac {2}{9} a A \tan (c+d x) \sec ^3(c+d x)+\frac {2 \left (49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+72 b B \sin (c+d x) a+3 A b^2 \sin (c+d x)\right ) \sec ^2(c+d x)}{315 a}+\frac {2 \left (75 B \sin (c+d x) a^3+88 A b \sin (c+d x) a^2+126 b C \sin (c+d x) a^2+9 b^2 B \sin (c+d x) a-4 A b^3 \sin (c+d x)\right ) \sec (c+d x)}{315 a^2}+\frac {2 \left (147 A a^4+189 C a^4+246 b B a^3+33 A b^2 a^2+63 b^2 C a^2-18 b^3 B a+8 A b^4\right ) \sin (c+d x)}{315 a^3}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 1.05, size = 5964, normalized size = 10.11 \[ \text {output too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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