Optimal. Leaf size=109 \[ \frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )}{d \sqrt{3 \sec (c+d x)-2}}-\frac{\sqrt{3 \sec (c+d x)-2} E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.175671, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3862, 3856, 2653, 3858, 2661} \[ \frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3 \sec (c+d x)-2}}-\frac{\sqrt{3 \sec (c+d x)-2} E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3862
Rule 3856
Rule 2653
Rule 3858
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-2+3 \sec (c+d x)}} \, dx &=-\left (\frac{1}{2} \int \frac{\sqrt{-2+3 \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\right )+\frac{3}{2} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2+3 \sec (c+d x)}} \, dx\\ &=\frac{\left (3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{3-2 \cos (c+d x)}} \, dx}{2 \sqrt{-2+3 \sec (c+d x)}}-\frac{\sqrt{-2+3 \sec (c+d x)} \int \sqrt{3-2 \cos (c+d x)} \, dx}{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{3 \sqrt{3-2 \cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right ) \sqrt{\sec (c+d x)}}{d \sqrt{-2+3 \sec (c+d x)}}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |-4\right ) \sqrt{-2+3 \sec (c+d x)}}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.117922, size = 68, normalized size = 0.62 \[ -\frac{\sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \left (E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )-3 \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )\right )}{d \sqrt{3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.282, size = 374, normalized size = 3.4 \begin{align*}{\frac{1}{2\,d\sin \left ( dx+c \right ) \left ( -3+2\,\cos \left ( dx+c \right ) \right ) } \left ( -2\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-2\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-10\,\cos \left ( dx+c \right ) +6 \right ) \sqrt{-{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}}}} \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{1}{\sqrt{3 \, \sec \left (d x + c\right ) - 2} \sqrt{\sec \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{\sqrt{3 \, \sec \left (d x + c\right ) - 2} \sqrt{\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right )^{2} - 2 \, \sec \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \sec{\left (c + d x \right )} - 2} \sqrt{\sec{\left (c + d x \right )}}}\, dx \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{1}{\sqrt{3 \, \sec \left (d x + c\right ) - 2} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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