Optimal. Leaf size=130 \[ \frac{2 B \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{a d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.155044, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {21, 4222, 2816} \[ \frac{2 B \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{a d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 4222
Rule 2816
Rubi steps
\begin{align*} \int \frac{(a B+b B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx &=B \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx\\ &=\left (B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 \sqrt{a+b} B \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{a d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.134802, size = 104, normalized size = 0.8 \[ \frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )}{d \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.679, size = 126, normalized size = 1. \begin{align*} 2\,{\frac{B\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d\sqrt{a+b\cos \left ( dx+c \right ) } \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{a+b\cos \left ( dx+c \right ) }{ \left ( a+b \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) }}}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) } \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 (B b \cos \left (d x + c\right ) + B a\right )} \sqrt{\sec \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{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{B \sqrt{\sec \left (d x + c\right )}}{\sqrt{b \cos \left (d x + c\right ) + a}}, 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 (B b \cos \left (d x + c\right ) + B a\right )} \sqrt{\sec \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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