Optimal. Leaf size=226 \[ -\frac{(8 A+9 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.749253, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {3046, 2983, 2982, 2782, 205, 2774, 216} \[ -\frac{(8 A+9 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2983
Rule 2982
Rule 2782
Rule 205
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx &=\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{2} a (6 A+5 C)-\frac{1}{2} a C \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\sqrt{\cos (c+d x)} \left (-\frac{3 a^2 C}{4}+\frac{3}{4} a^2 (8 A+7 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{6 a^2}\\ &=\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{8 d \sqrt{a+a \cos (c+d x)}}-\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\frac{3}{8} a^3 (8 A+7 C)-\frac{3}{8} a^3 (8 A+9 C) \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{6 a^3}\\ &=\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{8 d \sqrt{a+a \cos (c+d x)}}-\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+(A+C) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx-\frac{(8 A+9 C) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{16 a}\\ &=\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{8 d \sqrt{a+a \cos (c+d x)}}-\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}-\frac{(2 a (A+C)) \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 )}{d}+\frac{(8 A+9 C) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{8 a d}\\ &=-\frac{(8 A+9 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{8 \sqrt{a} d}+\frac{\sqrt{2} (A+C) \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 )}{\sqrt{a} d}+\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{8 d \sqrt{a+a \cos (c+d x)}}-\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 d \sqrt{a+a \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.10126, size = 349, normalized size = 1.54 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \left (4 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (24 A-2 C \cos (c+d x)+4 C \cos (2 (c+d x))+25 C)-\frac{3 i \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (16 \sqrt{2} (A+C) \log \left (1+e^{i (c+d x)}\right )-(8 A+9 C) \sinh ^{-1}\left (e^{i (c+d x)}\right )+8 A \log \left (1+\sqrt{1+e^{2 i (c+d x)}}\right )-16 \sqrt{2} A \log \left (\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right )-8 i A d x+9 C \log \left (1+\sqrt{1+e^{2 i (c+d x)}}\right )-16 \sqrt{2} C \log \left (\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right )-9 i C d x\right )}{\sqrt{1+e^{2 i (c+d x)}}}\right )}{48 d \sqrt{a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.126, size = 429, normalized size = 1.9 \begin{align*} -{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) ^{4}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}a} \left ( -24\,A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-48\,A\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-24\,A\sin \left ( dx+c \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-8\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+2\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+24\,A\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 ) +24\,C\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 ) -21\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+24\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ) +27\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ({\frac{\cos \left ( dx+c \right ) }{1+\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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 29.4987, size = 555, normalized size = 2.46 \begin{align*} \frac{{\left (8 \, C \cos \left (d x + c\right )^{2} - 2 \, C \cos \left (d x + c\right ) + 24 \, A + 21 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \,{\left ({\left (8 \, A + 9 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 9 \, C\right )} \sqrt{a} \arctan \left (\frac{\sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right ) - \frac{24 \, \sqrt{2}{\left ({\left (A + C\right )} a \cos \left (d x + c\right ) +{\left (A + C\right )} a\right )} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}}}{24 \,{\left (a d \cos \left (d x + c\right ) + a 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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{3}{2}}}{\sqrt{a \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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