Optimal. Leaf size=123 \[ -\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}-\frac{\sqrt{5} \sqrt{-3 \sec (c+d x)-2} E\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.204202, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3862, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}-\frac{\sqrt{5} \sqrt{-3 \sec (c+d x)-2} E\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3862
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-3 \sec (c+d x)} \sqrt{\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{\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{\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{\left (\sqrt{5} \sqrt{-2-3 \sec (c+d x)}\right ) \int \sqrt{\frac{3}{5}+\frac{2}{5} \cos (c+d x)} \, dx}{2 \sqrt{3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (3 \sqrt{3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{3}{5}+\frac{2}{5} \cos (c+d x)}} \, dx}{2 \sqrt{5} \sqrt{-2-3 \sec (c+d x)}}\\ &=-\frac{\sqrt{5} E\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right ) \sqrt{-2-3 \sec (c+d x)}}{d \sqrt{3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{3 \sqrt{3+2 \cos (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right ) \sqrt{\sec (c+d x)}}{\sqrt{5} d \sqrt{-2-3 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0886176, size = 78, normalized size = 0.63 \[ \frac{\sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \left (5 E\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )-3 \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.284, size = 390, normalized size = 3.2 \begin{align*} -{\frac{1}{10\,d\sin \left ( dx+c \right ) \left ( 3+2\,\cos \left ( dx+c \right ) \right ) } \left ( 2\,i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}-5\,i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}+2\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -5\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -20\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-10\,\cos \left ( dx+c \right ) +30 \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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