Optimal. Leaf size=126 \[ \frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac{7 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right )}{4 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{\cos (c+d x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.272263, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2778, 2983, 2982, 2781, 216, 2774} \[ \frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac{7 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right )}{4 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{\cos (c+d x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2778
Rule 2983
Rule 2982
Rule 2781
Rule 216
Rule 2774
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx &=\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\cos (c+d x)}}-\frac{1}{4} \int \frac{(-3+\cos (c+d x)) \sqrt{\cos (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{1+\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\cos (c+d x)}}-\frac{1}{4} \int \frac{\frac{1}{2}-\frac{7}{2} \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{1+\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\cos (c+d x)}}+\frac{7}{8} \int \frac{\sqrt{1+\cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx-\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{1+\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\cos (c+d x)}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,-\frac{\sin (c+d x)}{\sqrt{1+\cos (c+d x)}}\right )}{4 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{7 \sin ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1+\cos (c+d x)}}\right )}{4 d}-\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{1+\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1+\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.824201, size = 286, normalized size = 2.27 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left (-4 \sqrt{1+e^{2 i (c+d x)}} \sin \left (\frac{1}{2} (c+d x)\right )+2 \sqrt{1+e^{2 i (c+d x)}} \sin \left (\frac{3}{2} (c+d x)\right )-7 \sin \left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )+7 i \cos \left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )+7 \sinh ^{-1}\left (e^{i (c+d x)}\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )-i \cos \left (\frac{1}{2} (c+d x)\right )\right )+8 \sqrt{2} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )-i \cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{4 d \sqrt{1+e^{2 i (c+d x)}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.344, size = 187, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\sqrt{2+2\,\cos \left ( dx+c \right ) } \left ( 2\,\sin \left ( dx+c \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) +4\,\sqrt{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -\sin \left ( dx+c \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+7\,\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 ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{5}{2}}}{\sqrt{\cos \left (d x + c\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.11419, size = 402, normalized size = 3.19 \begin{align*} \frac{{\left (2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt{\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 4 \,{\left (\sqrt{2} \cos \left (d x + c\right ) + \sqrt{2}\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 )}\right ) - 7 \,{\left (\cos \left (d x + c\right ) + 1\right )} \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 )}{4 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\cos \left (d x + c\right )^{\frac{5}{2}}}{\sqrt{\cos \left (d x + c\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]