Optimal. Leaf size=177 \[ \frac{67 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.403644, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2765, 2977, 2978, 12, 2782, 205} \[ \frac{67 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2978
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{\int \frac{\sqrt{\cos (c+d x)} \left (\frac{3 a}{2}-5 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{13 \sqrt{\cos (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac{\int \frac{\frac{13 a^2}{4}-\frac{27}{2} a^2 \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{13 \sqrt{\cos (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac{67 \sqrt{\cos (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac{\int -\frac{15 a^3}{8 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{13 \sqrt{\cos (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac{67 \sqrt{\cos (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac{5 \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{13 \sqrt{\cos (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac{67 \sqrt{\cos (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{64 a^2 d}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac{13 \sqrt{\cos (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac{67 \sqrt{\cos (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.68773, size = 176, normalized size = 0.99 \[ \frac{\cos ^7\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (15 \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} \sin ^{-1}\left (\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )}}\right )+\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left (8 \tan ^4\left (\frac{1}{2} (c+d x)\right )-26 \tan ^2\left (\frac{1}{2} (c+d x)\right )+33\right )\right )}{24 a^4 d \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} (\cos (c+d x)+1)^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.378, size = 280, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{5}}{384\,d{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{11}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( 67\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+15\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -17\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+30\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -35\,\sqrt{2}\cos \left ( dx+c \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+15\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) -15\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3279, size = 585, normalized size = 3.31 \begin{align*} \frac{15 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \,{\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) + 2 \, \sqrt{a \cos \left (d x + c\right ) + a}{\left (67 \, \cos \left (d x + c\right )^{2} + 50 \, \cos \left (d x + c\right ) + 15\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\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{\cos \left (d x + c\right )^{\frac{5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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