Optimal. Leaf size=498 \[ \frac{\sqrt{a+b} \left (a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.991045, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4222, 2821, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{\sqrt{a+b} \left (a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{4 b d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4222
Rule 2821
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \cos (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} \, dx\\ &=\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{a b}{2}+b^2 \cos (c+d x)+\frac{1}{2} a b \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{2 b}\\ &=\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{a^2 b}{2}+a b^2 \cos (c+d x)-\frac{1}{2} b \left (a^2-4 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{4 b^2}\\ &=\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{a^2 b}{2}+a b^2 \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{4 b^2}-\frac{\left (\left (a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{8 b}\\ &=\frac{\sqrt{a+b} \left (a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\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}}}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b d}-\frac{\left (a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{8 b}+\frac{\left (a (a+2 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}{8 b}\\ &=-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) E\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}}}{4 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 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}}}{4 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (a^2-4 b^2\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\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}}}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b d}\\ \end{align*}
Mathematica [C] time = 18.0463, size = 1113, normalized size = 2.23 \[ \frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{a^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left (\frac{1}{2} (c+d x)\right )-a b \sqrt{\frac{a-b}{a+b}} \tan ^5\left (\frac{1}{2} (c+d x)\right )+2 a b \sqrt{\frac{a-b}{a+b}} \tan ^3\left (\frac{1}{2} (c+d x)\right )-2 i a^2 \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \tan ^2\left (\frac{1}{2} (c+d x)\right )+8 i b^2 \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \tan ^2\left (\frac{1}{2} (c+d x)\right )-a^2 \sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )-a b \sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )-i a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}+2 i \left (a^2+b a-2 b^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}-2 i a^2 \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}+8 i b^2 \Pi \left (\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{a-b}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}}{4 b \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt{\frac{a \tan ^2\left (\frac{1}{2} (c+d x)\right )-b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.546, size = 1241, normalized size = 2.5 \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{b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\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{\sqrt{b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{3}{2}}}, 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{a + b \cos{\left (c + d x \right )}}}{\sec ^{\frac{3}{2}}{\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{\sqrt{b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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