Optimal. Leaf size=126 \[ \frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {\cos (c+d x)+1}}-\frac {\sqrt {2} \sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac {7 \sin ^{-1}\left (\frac {\sin (c+d x)}{\sqrt {\cos (c+d x)+1}}\right )}{4 d}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{4 d \sqrt {\cos (c+d x)+1}} \]
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Rubi [A] time = 0.27, antiderivative size = 126, normalized size of antiderivative = 1.00, 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.
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Rule 216
Rule 2774
Rule 2778
Rule 2781
Rule 2982
Rule 2983
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*}
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Mathematica [C] time = 0.90, 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.
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fricas [A] time = 1.99, size = 135, normalized size = 1.07 \[ \frac {{\left (2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 4 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) - 7 \, {\left (\cos \left (d x + c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\cos \left (d x + c\right ) + 1} \sqrt {\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right )}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right ) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 187, normalized size = 1.48 \[ -\frac {\sqrt {2+2 \cos \left (d x +c \right )}\, \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{3} \left (2 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+4 \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+7 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )\right ) \sqrt {2}}{8 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right ) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^{5/2}}{\sqrt {\cos \left (c+d\,x\right )+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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