Optimal. Leaf size=104 \[ \frac{F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{3 d}-\frac{4 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d} \]
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Rubi [A] time = 0.250712, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2802, 3060, 2654, 3002, 2662, 2806} \[ \frac{F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{3 d}-\frac{4 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2802
Rule 3060
Rule 2654
Rule 3002
Rule 2662
Rule 2806
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx &=\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} \int \frac{\left (2+2 \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{12} \int \frac{(8+6 \cos (c+d x)) \sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx-\frac{1}{6} \int \sqrt{3-4 \cos (c+d x)} \, dx\\ &=-\frac{\sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{2} \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx+\frac{2}{3} \int \frac{\sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{3 d}+\frac{F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{4 \Pi \left (2;\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 1.4186, size = 179, normalized size = 1.72 \[ \frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)+\frac{6 \sqrt{4 \cos (c+d x)-3} \Pi \left (2;\left .\frac{1}{2} (c+d x)\right |8\right )}{\sqrt{3-4 \cos (c+d x)}}-\frac{i \sin (c+d x) \left (-12 F\left (i \sinh ^{-1}\left (\sqrt{3-4 \cos (c+d x)}\right )|-\frac{1}{7}\right )+21 E\left (i \sinh ^{-1}\left (\sqrt{3-4 \cos (c+d x)}\right )|-\frac{1}{7}\right )-8 \Pi \left (-\frac{1}{3};i \sinh ^{-1}\left (\sqrt{3-4 \cos (c+d x)}\right )|-\frac{1}{7}\right )\right )}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.337, size = 351, normalized size = 3.4 \begin{align*} -{\frac{1}{d}\sqrt{- \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -{\frac{2}{3}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1}}+{\frac{1}{7}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,{\frac{2\,\sqrt{14}}{7}} \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}}-{\frac{1}{3}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,{\frac{2\,\sqrt{14}}{7}} \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}}-{\frac{4}{21}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticPi} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,2,{\frac{2\,\sqrt{14}}{7}} \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+7}}}} \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{\sec \left (d x + c\right )^{2}}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{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{-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2}}{4 \, \cos \left (d x + c\right ) - 3}, 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{\sec ^{2}{\left (c + d x \right )}}{\sqrt{3 - 4 \cos{\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{\sec \left (d x + c\right )^{2}}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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