Optimal. Leaf size=462 \[ -\frac{2 \left (-8 a^2 C+22 a b B-81 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left (110 a^2 b B-40 a^3 C-335 a b^2 C-539 b^3 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left (-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-3705 a b^4 C-1617 b^5 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]
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Rubi [A] time = 0.979887, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.225, Rules used = {3029, 2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (-8 a^2 C+22 a b B-81 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left (110 a^2 b B-40 a^3 C-335 a b^2 C-539 b^3 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left (-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (-285 a^2 b^2 C+110 a^3 b B-40 a^4 C-1254 a b^3 B-675 b^4 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-3705 a b^4 C-1617 b^5 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2990
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \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 \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, 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{9}{2} b C \cos (c+d x)+\frac{1}{2} (11 b B-4 a C) \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac{2 (11 b B-4 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{1}{4} b (77 b B-10 a C)-\frac{1}{4} \left (22 a b B-8 a^2 C-81 b^2 C\right ) \cos (c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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{3}{8} b \left (143 a b B-10 a^2 C+135 b^2 C\right )-\frac{1}{8} \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac{2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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{3}{16} b \left (605 a^2 b B+539 b^3 B-10 a^3 C+1010 a b^2 C\right )-\frac{3}{16} \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \cos (c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac{2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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{3}{32} b \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+3315 a^2 b^2 C+675 b^4 C\right )-\frac{3}{32} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac{2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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 (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3465 b^3}-\frac{\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3465 b^3}\\ &=-\frac{2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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 (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 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}{3465 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 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}{3465 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 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 )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 b B-4 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*}
Mathematica [A] time = 2.03145, size = 357, normalized size = 0.77 \[ \frac{b (a+b \cos (c+d x)) \left (\left (18660 a^2 b^2 C+880 a^3 b B-320 a^4 C+32868 a b^3 B+13050 b^4 C\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 b B+30 a^3 C+3095 a b^2 C+1463 b^3 B\right ) \sin (2 (c+d x))+5 b \left (\left (452 a^2 C+836 a b B+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((46 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (3315 a^2 b^2 C+1705 a^3 b B+10 a^4 C+2871 a b^3 B+675 b^4 C\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (3069 a^2 b^3 B+255 a^3 b^2 C-110 a^4 b B+40 a^5 C+3705 a b^4 C+1617 b^5 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{27720 b^3 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.177, size = 1983, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + B a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{b \cos \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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