Optimal. Leaf size=218 \[ -\frac{2 \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a C \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d} \]
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Rubi [A] time = 0.325578, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3024, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a C \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \sqrt{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{2 \int \sqrt{a+b \cos (c+d x)} \left (\frac{1}{2} b (5 A+3 C)-a C \cos (c+d x)\right ) \, dx}{5 b}\\ &=-\frac{4 a C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac{2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{4 \int \frac{\frac{1}{4} a b (15 A+7 C)-\frac{1}{4} \left (2 a^2 C-3 b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b}\\ &=-\frac{4 a C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac{2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{\left (2 a \left (a^2-b^2\right ) C\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^2}+\frac{1}{15} \left (15 A+\left (9-\frac{2 a^2}{b^2}\right ) C\right ) \int \sqrt{a+b \cos (c+d x)} \, dx\\ &=-\frac{4 a C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac{2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}+\frac{\left (\left (15 A+\left (9-\frac{2 a^2}{b^2}\right ) 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}{15 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (2 a \left (a^2-b^2\right ) C \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}{15 b^2 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (15 A+\left (9-\frac{2 a^2}{b^2}\right ) C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a \left (a^2-b^2\right ) C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac{2 C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [A] time = 0.92953, size = 181, normalized size = 0.83 \[ \frac{-2 (a+b) \left (2 a^2 C-15 A b^2-9 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+b C \sin (c+d x) \left (2 a^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right )+4 a C \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.355, size = 821, normalized size = 3.8 \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} + A\right )} \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 ({\left (C \cos \left (d x + c\right )^{2} + A\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(-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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