Optimal. Leaf size=174 \[ \frac{2 \left (2 a^2 C+b^2 (3 A+C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d} \]
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Rubi [A] time = 0.205793, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3024, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (2 a^2 C+b^2 (3 A+C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx &=\frac{2 C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b d}+\frac{2 \int \frac{\frac{1}{2} b (3 A+C)-a C \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 b}\\ &=\frac{2 C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b d}-\frac{(2 a C) \int \sqrt{a+b \cos (c+d x)} \, dx}{3 b^2}+\frac{1}{3} \left (3 A+C+\frac{2 a^2 C}{b^2}\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b d}-\frac{\left (2 a C \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{3 b^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (3 A+C+\frac{2 a^2 C}{b^2}\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}{3 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (3 A+C+\frac{2 a^2 C}{b^2}\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 )}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.683545, size = 148, normalized size = 0.85 \[ \frac{2 \left (C \left (2 a^2+b^2\right )+3 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+2 b C \sin (c+d x) (a+b \cos (c+d x))-4 a C (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.378, size = 532, normalized size = 3.1 \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{C \cos \left (d x + c\right )^{2} + A}{\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 )^{2} + A}{\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{C \cos \left (d x + c\right )^{2} + A}{\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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