Optimal. Leaf size=523 \[ \frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac {4 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{45045 b^3 d}-\frac {2 \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]
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Rubi [A] time = 1.28, antiderivative size = 523, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {3050, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}-\frac {2 \left (10 a^2 b^2 (143 A+124 C)+240 a^4 C-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}-\frac {4 a \left (5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{45045 b^3 d}+\frac {4 a \left (a^2-b^2\right ) \left (5 a^2 b^2 (143 A+94 C)+120 a^4 C-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 C-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3023
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {2 \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (2 a C+\frac {1}{2} b (13 A+11 C) \cos (c+d x)-3 a C \cos ^2(c+d x)\right ) \, dx}{13 b}\\ &=-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (-3 a^2 C-\frac {5}{2} a b C \cos (c+d x)+\frac {1}{4} \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx}{143 b^2}\\ &=\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {8 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{8} b \left (1001 A b^2+60 a^2 C+847 b^2 C\right )-\frac {1}{4} a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) \cos (c+d x)\right ) \, dx}{1287 b^3}\\ &=-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {16 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{16} a b \left (1859 A b^2+60 a^2 C+1423 b^2 C\right )-\frac {1}{16} \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) \cos (c+d x)\right ) \, dx}{9009 b^3}\\ &=-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {32 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{32} b \left (60 a^4 C+539 b^4 (13 A+11 C)+5 a^2 b^2 (1573 A+1175 C)\right )-\frac {3}{16} a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right ) \, dx}{45045 b^3}\\ &=-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {64 \int \frac {-\frac {3}{64} a b \left (60 a^4 C-5 a^2 b^2 (4433 A+3337 C)-3 b^4 (12441 A+10277 C)\right )-\frac {3}{64} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{135135 b^3}\\ &=-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {\left (2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{45045 b^4}-\frac {\left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{45045 b^4}\\ &=-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}-\frac {\left (\left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{45045 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{45045 b^4 \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {2 \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}\\ \end {align*}
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Mathematica [A] time = 2.59, size = 395, normalized size = 0.76 \[ \frac {b (a+b \cos (c+d x)) \left (4 a \left (960 a^4 C+10 a^2 b^2 (572 A+331 C)+3 b^4 (71214 A+60793 C)\right ) \sin (c+d x)+b \left (5 b \left (2 a \left (60 a^2 C+10868 A b^2+13939 b^2 C\right ) \sin (3 (c+d x))+7 b \left (\left (636 a^2 C+572 A b^2+880 b^2 C\right ) \sin (4 (c+d x))+9 b C (54 a \sin (5 (c+d x))+11 b \sin (6 (c+d x)))\right )\right )+\left (-1440 a^4 C+120 a^2 b^2 (1430 A+1457 C)+77 b^4 (1976 A+1897 C)\right ) \sin (2 (c+d x))\right )\right )+32 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (a b^2 \left (-60 a^4 C+5 a^2 b^2 (4433 A+3337 C)+3 b^4 (12441 A+10277 C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{720720 b^4 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{6} + 2 \, C a b \cos \left (d x + c\right )^{5} + 2 \, A a b \cos \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.71, size = 2223, normalized size = 4.25 \[ \text {Expression too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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