Optimal. Leaf size=78 \[ \frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 A b^2 \tan (c+d x) \sqrt{b \sec (c+d x)}}{3 d} \]
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Rubi [A] time = 0.0957784, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3238, 4046, 3771, 2641} \[ \frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 A b^2 \tan (c+d x) \sqrt{b \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3238
Rule 4046
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/2} \, dx &=b^2 \int \sqrt{b \sec (c+d x)} \left (C+A \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 A b^2 \sqrt{b \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} \left (b^2 (A+3 C)\right ) \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 A b^2 \sqrt{b \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} \left (b^2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 A b^2 \sqrt{b \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.221161, size = 58, normalized size = 0.74 \[ \frac{2 b^2 \sqrt{b \sec (c+d x)} \left ((A+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+A \tan (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.587, size = 199, normalized size = 2.6 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( iA\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( 1+\cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) +3\,iC\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{ \left ( 1+\cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -A\cos \left ( dx+c \right ) +A \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}}} \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 )} \left (b \sec \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 ({\left (C b^{2} \cos \left (d x + c\right )^{2} + A b^{2}\right )} \sqrt{b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \sec \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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