Optimal. Leaf size=580 \[ \frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{105 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-\left (a^3 (25 A-63 B+35 C)\right )+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a^2 d}+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 b^2 C \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{d} \]
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Rubi [A] time = 1.80, antiderivative size = 580, 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 = {3047, 3053, 2809, 2998, 2816, 2994} \[ \frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{105 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {a+b} \cot (c+d x) \left (a^2 b (145 A-119 B+245 C)+a^3 (-(25 A-63 B+35 C))-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (5 a^2 b (29 A+49 C)+63 a^3 B+161 a b^2 B+15 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a^2 d}+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 b^2 C \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2994
Rule 2998
Rule 3047
Rule 3053
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+7 a B)+\frac {1}{2} (5 a A+7 b B+7 a C) \cos (c+d x)+\frac {7}{2} b C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4}{35} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} \left (40 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \cos (c+d x)+\frac {35}{4} b^2 C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8}{105} \int \frac {\frac {1}{8} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right )+\frac {1}{8} \left (119 a^2 b B+105 b^3 B+45 a b^2 (3 A+7 C)+5 a^3 (5 A+7 C)\right ) \cos (c+d x)+\frac {105}{8} b^3 C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8}{105} \int \frac {\frac {1}{8} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right )+\frac {1}{8} \left (119 a^2 b B+105 b^3 B+45 a b^2 (3 A+7 C)+5 a^3 (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\left (b^3 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \sqrt {a+b} C \cot (c+d x) \Pi \left (\frac {a+b}{b};\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}}}{d}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{105} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\frac {1}{105} \left (-15 b^3 (A-7 B)+a^3 (25 A-63 B+35 C)-a^2 b (145 A-119 B+245 C)+a b^2 (135 A-161 B+315 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 (a-b) \sqrt {a+b} \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) \cot (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}}}{105 a^2 d}-\frac {2 \sqrt {a+b} \left (15 b^3 (A-7 B)-a^3 (25 A-63 B+35 C)+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)\right ) \cot (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}}}{105 a d}-\frac {2 b^2 \sqrt {a+b} C \cot (c+d x) \Pi \left (\frac {a+b}{b};\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}}}{d}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+7 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] time = 6.96, size = 1472, normalized size = 2.54 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C b^{2} \cos \left (d x + c\right )^{4} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]
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 = 0.73, size = 5143, normalized size = 8.87 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^{9/2}} \,d x \]
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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