Optimal. Leaf size=87 \[ \frac{2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.101688, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3021, 2748, 2636, 2639, 2641} \[ \frac{2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{\frac{3 B}{2}+\frac{1}{2} (A+3 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+B \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{3} (A+3 C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-B \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.522094, size = 69, normalized size = 0.79 \[ \frac{\frac{2 \sin (c+d x) (A+3 B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-6 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.376, size = 500, normalized size = 5.8 \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 \frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{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} + B \cos \left (d x + c\right ) + A}{\cos \left (d x + c\right )^{\frac{5}{2}}}, 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} + B \cos \left (d x + c\right ) + A}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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