Optimal. Leaf size=62 \[ \frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0845289, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2779, 2781, 216} \[ \frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2779
Rule 2781
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx &=\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}}-\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx\\ &=\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}}+\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 (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.7732, size = 178, normalized size = 2.87 \[ \frac{2 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (\frac{1}{2} \cos (c+d x) (\cos (c+d x)+2) \csc ^4\left (\frac{1}{2} (c+d x)\right ) \left (-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left (\sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right ) (-\sec (c+d x))}\right )+1\right )-\frac{1}{10} \sin (c+d x) \tan (c+d x) \text{Hypergeometric2F1}\left (2,\frac{5}{2},\frac{7}{2},\sin ^2\left (\frac{1}{2} (c+d x)\right ) (-\sec (c+d x))\right )\right )}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.289, size = 210, normalized size = 3.4 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ( \sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}+2\,\sqrt{2}\cos \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+\sqrt{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}+2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) \sqrt{2+2\,\cos \left ( dx+c \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.13109, size = 343, normalized size = 5.53 \begin{align*} -\frac{{\left (\sqrt{2} \cos \left (d x + c\right )^{2} + \sqrt{2} \cos \left (d x + c\right )\right )} \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 )}{2 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )}}\right ) - 2 \, \sqrt{\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos{\left (c + d x \right )} + 1} \cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cos \left (d x + c\right ) + 1} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]