Optimal. Leaf size=378 \[ \frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^4 d}-\frac{2 a \left (4 a^2 b^2 (42 A+19 C)+128 a^4 C+3 b^4 (49 A+37 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^5 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)}}{9 b d} \]
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Rubi [A] time = 0.926682, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3050, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^4 d}-\frac{2 a \left (4 a^2 b^2 (42 A+19 C)+128 a^4 C+3 b^4 (49 A+37 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^5 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)}}{9 b d} \]
Antiderivative was successfully verified.
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Rule 3050
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx &=\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{2 \int \frac{\cos ^2(c+d x) \left (3 a C+\frac{1}{2} b (9 A+7 C) \cos (c+d x)-4 a C \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{9 b}\\ &=-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{4 \int \frac{\cos (c+d x) \left (-8 a^2 C+\frac{1}{2} a b C \cos (c+d x)+\frac{1}{4} \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{63 b^2}\\ &=\frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{8 \int \frac{\frac{1}{4} a \left (48 a^2 C+7 b^2 (9 A+7 C)\right )+\frac{1}{8} b \left (189 A b^2-16 a^2 C+147 b^2 C\right ) \cos (c+d x)-\frac{3}{4} a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^3}\\ &=-\frac{4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{16 \int \frac{\frac{3}{8} a b \left (21 A b^2+16 a^2 C+18 b^2 C\right )+\frac{3}{16} \left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{945 b^4}\\ &=-\frac{4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{\left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{315 b^5}-\frac{\left (a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^5}\\ &=-\frac{4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}+\frac{\left (\left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 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}{315 b^5 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 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}{315 b^5 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 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 )}{315 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac{2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{16 a C \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{9 b d}\\ \end{align*}
Mathematica [A] time = 1.39748, size = 272, normalized size = 0.72 \[ \frac{8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b \left (32 a^3 b C+6 a b^3 (7 A+6 C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (12 a^2 b^2 (14 A+9 C)+128 a^4 C+21 b^4 (9 A+7 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (32 a \left (2 C \left (8 a^2+9 b^2\right )+21 A b^2\right ) \sin (c+d x)-b \left (2 \left (96 a^2 C+126 A b^2+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (7 b \sin (4 (c+d x))-16 a \sin (3 (c+d x)))\right )\right )}{1260 b^5 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.422, size = 1527, normalized size = 4. \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{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 (\frac{C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}}{\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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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