Integrand size = 44, antiderivative size = 779 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (45 B+64 C)-8 a b^3 (355 B+193 C)-4 a^2 b^2 (295 B+423 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d} \]
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Time = 3.52 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {3108, 3069, 3128, 3140, 3132, 2888, 3077, 2895, 3073} \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (-15 a^2 C+50 a b B+64 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac {\left (-15 a^3 C+50 a^2 b B+172 a b^2 C+120 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{320 b d}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (295 B+423 C)-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{1920 b^2 d}-\frac {(a-b) \sqrt {a+b} \left (-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{1920 a b^2 d}+\frac {\left (-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\sqrt {a+b} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 C-240 a^2 b^3 B-240 a b^4 C-96 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{128 b^3 d}+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d} \]
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Rule 2888
Rule 2895
Rule 3069
Rule 3073
Rule 3077
Rule 3108
Rule 3128
Rule 3132
Rule 3140
Rubi steps \begin{align*} \text {integral}& = \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, dx \\ & = \frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left (\frac {a C}{2}+4 b C \cos (c+d x)+\frac {1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{5 b} \\ & = \frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {5}{4} a (2 b B+a C)+\frac {3}{2} b (10 b B+9 a C) \cos (c+d x)+\frac {1}{4} \left (50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{20 b} \\ & = \frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{8} a \left (110 a b B+15 a^2 C+64 b^2 C\right )+\frac {1}{4} b \left (310 a b B+147 a^2 C+128 b^2 C\right ) \cos (c+d x)+\frac {3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{60 b} \\ & = \frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {\frac {1}{16} a \left (590 a^2 b B+360 b^3 B+15 a^3 C+772 a b^2 C\right )+\frac {1}{8} b \left (1610 a^2 b B+360 b^3 B+573 a^3 C+1156 a b^2 C\right ) \cos (c+d x)+\frac {1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{120 b} \\ & = \frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {-\frac {1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+772 a b^2 C\right ) \cos (c+d x)-\frac {15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2} \\ & = \frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac {\int \frac {-\frac {1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right )+\frac {1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+772 a b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{240 b^2}-\frac {\left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 b^2} \\ & = \frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}-\frac {\left (a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3840 b^2}-\frac {\left (a \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (45 B+64 C)-8 a b^3 (355 B+193 C)-4 a^2 b^2 (295 B+423 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3840 b^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \cot (c+d x) E\left (\arcsin \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}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (45 B+64 C)-8 a b^3 (355 B+193 C)-4 a^2 b^2 (295 B+423 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \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}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 C-240 a b^4 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+1692 a^2 b^2 C+1024 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+172 a b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.87 (sec) , antiderivative size = 1353, normalized size of antiderivative = 1.74 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {-\frac {4 a \left (-1330 a^3 b B-3560 a b^3 B+15 a^4 C-3236 a^2 b^2 C-1024 b^4 C\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (-6440 a^2 b^2 B-1440 b^4 B-2292 a^3 b C-4624 a b^3 C\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (-150 a^3 b B-2840 a b^3 B+45 a^4 C-1692 a^2 b^2 C-1024 b^4 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{3840 b d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {\left (590 a^2 b B+420 b^3 B+15 a^3 C+898 a b^2 C\right ) \sin (c+d x)}{960 b}+\frac {1}{480} \left (170 a b B+93 a^2 C+88 b^2 C\right ) \sin (2 (c+d x))+\frac {1}{160} b (10 b B+21 a C) \sin (3 (c+d x))+\frac {1}{40} b^2 C \sin (4 (c+d x))\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(7109\) vs. \(2(719)=1438\).
Time = 10.21 (sec) , antiderivative size = 7110, normalized size of antiderivative = 9.13
method | result | size |
parts | \(\text {Expression too large to display}\) | \(7110\) |
default | \(\text {Expression too large to display}\) | \(7197\) |
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Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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