Optimal. Leaf size=116 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (3 A+C)+b B)}{3 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d} \]
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Rubi [A] time = 0.217469, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3033, 3023, 2748, 2641, 2639} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a (3 A+C)+b B)}{3 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3033
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2}{5} \int \frac{\frac{5 a A}{2}+\frac{1}{2} (5 A b+5 a B+3 b C) \cos (c+d x)+\frac{5}{2} (b B+a C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (b B+a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{4}{15} \int \frac{\frac{5}{4} (b B+a (3 A+C))+\frac{3}{4} (5 A b+5 a B+3 b C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (b B+a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{1}{5} (5 A b+5 a B+3 b C) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} (b B+a (3 A+C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (5 A b+5 a B+3 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 (b B+a (3 A+C)) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 (b B+a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.593086, size = 94, normalized size = 0.81 \[ \frac{2 \left (5 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (3 a A+a C+b B)+3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+3 b C)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 a C+5 b B+3 b C \cos (c+d x))\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.844, size = 465, 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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}}{\sqrt{\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 (\frac{C b \cos \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}, 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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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