Optimal. Leaf size=118 \[ \frac{2 \sqrt{\sec (d+e x)} \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} \text{EllipticF}\left (\frac{1}{2} \left (-\tan ^{-1}(a,c)+d+e x\right ),\frac{2 \sqrt{a^2+c^2}}{\sqrt{a^2+c^2}+b}\right )}{e \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \]
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Rubi [A] time = 0.166285, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3167, 3127, 2661} \[ \frac{2 \sqrt{\sec (d+e x)} \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \]
Antiderivative was successfully verified.
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Rule 3167
Rule 3127
Rule 2661
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (d+e x)}}{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \, dx &=\frac{\left (\sqrt{\sec (d+e x)} \sqrt{b+a \cos (d+e x)+c \sin (d+e x)}\right ) \int \frac{1}{\sqrt{b+a \cos (d+e x)+c \sin (d+e x)}} \, dx}{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}\\ &=\frac{\left (\sqrt{\sec (d+e x)} \sqrt{\frac{b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt{a^2+c^2}}}\right ) \int \frac{1}{\sqrt{\frac{b}{b+\sqrt{a^2+c^2}}+\frac{\sqrt{a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt{a^2+c^2}}}} \, dx}{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}\\ &=\frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right ) \sqrt{\sec (d+e x)} \sqrt{\frac{b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}{e \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}\\ \end{align*}
Mathematica [C] time = 0.91945, size = 339, normalized size = 2.87 \[ \frac{2 \sqrt{\sec (d+e x)} \sec \left (\tan ^{-1}\left (\frac{a}{c}\right )+d+e x\right ) \sqrt{-\frac{c \sqrt{\frac{a^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac{a}{c}\right )+d+e x\right )-1\right )}{c \sqrt{\frac{a^2}{c^2}+1}+b}} \sqrt{\frac{c \sqrt{\frac{a^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac{a}{c}\right )+d+e x\right )+1\right )}{c \sqrt{\frac{a^2}{c^2}+1}-b}} \sqrt{c \sqrt{\frac{a^2}{c^2}+1} \sin \left (\tan ^{-1}\left (\frac{a}{c}\right )+d+e x\right )+b} \sqrt{a \cos (d+e x)+b+c \sin (d+e x)} F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac{a}{c}\right )\right )}{b-\sqrt{\frac{a^2}{c^2}+1} c},\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac{a}{c}\right )\right )}{b+\sqrt{\frac{a^2}{c^2}+1} c}\right )}{c e \sqrt{\frac{a^2}{c^2}+1} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.596, size = 722, normalized size = 6.1 \begin{align*} \text{result too large to display} \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{\sqrt{\sec \left (e x + d\right )}}{\sqrt{b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a}}\,{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{\sec \left (e x + d\right )}}{\sqrt{b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a}}, 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{\sqrt{\sec{\left (d + e x \right )}}}{\sqrt{a + b \sec{\left (d + e x \right )} + c \tan{\left (d + e 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{\sqrt{\sec \left (e x + d\right )}}{\sqrt{b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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