Optimal. Leaf size=134 \[ -\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}} \]
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Rubi [A] time = 0.242472, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2779, 2984, 12, 2781, 216} \[ -\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )}{d}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}} \]
Antiderivative was successfully verified.
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Rule 2779
Rule 2984
Rule 12
Rule 2781
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{1}{5} \int \frac{1-4 \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2}{15} \int \frac{-\frac{13}{2}+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}}-\frac{4}{15} \int \frac{15}{4 \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 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)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 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)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}-\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.95131, size = 1538, normalized size = 11.48 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.316, size = 344, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{30\,d \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3} \left ( 1+\cos \left ( dx+c \right ) \right ) ^{4}} \left ( 15\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +60\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +90\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +60\,\sqrt{2}\cos \left ( dx+c \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +15\,\sqrt{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +26\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +6\,\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{7}{2}}}} \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 = 2.13795, size = 410, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (13 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 3\right )} \sqrt{\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \,{\left (\sqrt{2} \cos \left (d x + c\right )^{4} + \sqrt{2} \cos \left (d x + c\right )^{3}\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 )}{15 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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