Optimal. Leaf size=139 \[ \frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.202712, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {16, 3021, 2748, 2636, 2640, 2639, 2642, 2641} \[ \frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3021
Rule 2748
Rule 2636
Rule 2640
Rule 2639
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx &=b^2 \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 b \sin (c+d x)}{3 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}\\ &=\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+(b B) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx+\frac{1}{3} (A+3 C) \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx\\ &=\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{B \int \sqrt{b \cos (c+d x)} \, dx}{b}+\frac{\left ((A+3 C) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \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 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{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 \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 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 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.29658, size = 757, normalized size = 5.45 \[ \frac{2 B \csc (c) \cos ^{\frac{5}{2}}(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{d \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 A \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{3 d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 C \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac{4 \sec (c) \sec (c+d x) (A \sin (c)+3 B \sin (d x))}{3 d}+\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 B \csc (c) \sec (c)}{d}\right )}{\sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 8.132, size = 508, normalized size = 3.7 \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 )} \sec \left (d x + c\right )^{2}}{\sqrt{b \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{{\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 )} \sec \left (d x + c\right )^{2}}{b \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 )} \sec \left (d x + c\right )^{2}}{\sqrt{b \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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