Optimal. Leaf size=435 \[ \frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 b^2 d}-\frac {2 \left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]
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Rubi [A] time = 0.88, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac {2 \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 b^2 d}-\frac {2 \left (a^2-b^2\right ) \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 3023
Rule 3050
Rubi steps
\begin {align*} \int \cos (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 (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {2 \int (a+b \cos (c+d x))^{5/2} \left (a C+\frac {1}{2} b (11 A+9 C) \cos (c+d x)-2 a C \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (-\frac {5}{2} a b C+\frac {1}{4} \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{99 b^2}\\ &=\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {8 \int (a+b \cos (c+d x))^{3/2} \left (\frac {15}{8} b \left (33 A b^2-2 a^2 C+27 b^2 C\right )+\frac {5}{8} a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^2}\\ &=\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {16 \int \sqrt {a+b \cos (c+d x)} \left (\frac {15}{8} a b \left (132 A b^2-\left (a^2-101 b^2\right ) C\right )+\frac {15}{16} \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^2}\\ &=\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {32 \int \frac {\frac {15}{32} b \left (2 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (297 A+221 C)\right )+\frac {15}{32} a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^2}\\ &=\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}-\frac {\left (\left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{693 b^3}+\frac {\left (a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{693 b^3}\\ &=\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {\left (a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 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}{693 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 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}{693 b^3 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 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 )}{693 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 328, normalized size = 0.75 \[ \frac {b (a+b \cos (c+d x)) \left (b \left (4 a \left (6 a^2 C+594 A b^2+619 b^2 C\right ) \sin (2 (c+d x))+b \left (\left (452 a^2 C+396 A b^2+513 b^2 C\right ) \sin (3 (c+d x))+7 b C (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right )\right )+\left (-64 a^4 C+12 a^2 b^2 (396 A+311 C)+6 b^4 (506 A+435 C)\right ) \sin (c+d x)\right )+16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (2 a^4 b C+3 a^2 b^3 (297 A+221 C)+15 b^5 (11 A+9 C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{5544 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + 2 \, C a b \cos \left (d x + c\right )^{4} + 2 \, A a b \cos \left (d x + c\right )^{2} + A a^{2} \cos \left (d x + c\right ) + {\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.70, size = 1791, normalized size = 4.12 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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