Optimal. Leaf size=54 \[ \frac{2 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right )}{d}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \]
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Rubi [A] time = 0.116653, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2777, 2774, 216, 2781} \[ \frac{2 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right )}{d}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2777
Rule 2774
Rule 216
Rule 2781
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx &=-\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx+\int \frac{\sqrt{1+\cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,-\frac{\sin (c+d x)}{\sqrt{1+\cos (c+d x)}}\right )}{d}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,-\frac{\sin (c+d x)}{1+\cos (c+d x)}\right )}{d}\\ &=-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{1+\cos (c+d x)}\right )}{d}+\frac{2 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1+\cos (c+d x)}}\right )}{d}\\ \end{align*}
Mathematica [C] time = 0.247385, size = 135, normalized size = 2.5 \[ -\frac{i \left (1+e^{i (c+d x)}\right ) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left (\sinh ^{-1}\left (e^{i (c+d x)}\right )-\sqrt{2} \tanh ^{-1}\left (\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )-\tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{d \sqrt{1+e^{2 i (c+d x)}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.296, size = 124, normalized size = 2.3 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}\sqrt{2+2\,\cos \left ( dx+c \right ) }\sqrt{\cos \left ( dx+c \right ) } \left ( \sqrt{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +2\,\arctan \left ({\frac{\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \right ){\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9667, size = 204, normalized size = 3.78 \begin{align*} \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) - 2 \, \arctan \left (\frac{\sqrt{\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right )}{d} \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{\cos{\left (c + d x \right )}}}{\sqrt{\cos{\left (c + d x \right )} + 1}}\, 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{\cos \left (d x + c\right )}}{\sqrt{\cos \left (d x + c\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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