Optimal. Leaf size=147 \[ \frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.162703, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3021, 2748, 2636, 2640, 2639, 2642, 2641} \[ \frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3021
Rule 2748
Rule 2636
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 \int \frac{\frac{3 b^2 B}{2}+\frac{1}{2} b^2 (A+3 C) \cos (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx}{3 b^3}\\ &=\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{B \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx}{b}+\frac{(A+3 C) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{3 b^2}\\ &=\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \int \sqrt{b \cos (c+d x)} \, dx}{b^3}+\frac{\left ((A+3 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{\left (B \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b^3 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 B \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.377389, size = 92, normalized size = 0.63 \[ \frac{2 \left (\tan (c+d x) (A+3 B \cos (c+d x))+(A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-3 B \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.615, size = 508, normalized size = 3.5 \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}{\left (b \cos \left (d x + c\right )\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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{3} \cos \left (d x + c\right )^{3}}, 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}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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