Optimal. Leaf size=152 \[ -\frac{(7 A-C) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.37834, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {3042, 2984, 12, 2782, 205} \[ -\frac{(7 A-C) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\frac{1}{2} a (5 A+C)-a (A-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{\int -\frac{a^2 (7 A-C)}{4 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{a^3}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}-\frac{(7 A-C) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{(7 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 )}{2 d}\\ &=-\frac{(7 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 )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.83154, size = 434, normalized size = 2.86 \[ \frac{\cos ^3\left (\frac{1}{2} (c+d x)\right ) \left (\frac{(A-7 C) \csc ^3\left (\frac{1}{2} (c+d x)\right ) \left (5 (4 \cos (c+d x)+\cos (2 (c+d x))+1) \left (-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left (\sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right ) (-\sec (c+d x))}\right )+1\right )-2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sin (c+d x) \tan (c+d x) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )}{2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 (A+C) \left (2 \sin \left (\frac{1}{2} (c+d x)\right )-1\right )}{\sqrt{\cos (c+d x)} \left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )^2}-\frac{5 (A+C) \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+1\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )-1\right ) \sqrt{\cos (c+d x)}}-\frac{20 (A+C) \sqrt{\cos (c+d x)}}{\sin \left (\frac{1}{2} (c+d x)\right )-1}-\frac{20 (A+C) \sqrt{\cos (c+d x)}}{\sin \left (\frac{1}{2} (c+d x)\right )+1}+30 (A+C) \tan ^{-1}\left (\frac{1-2 \sin \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\cos (c+d x)}}\right )-30 (A+C) \tan ^{-1}\left (\frac{2 \sin \left (\frac{1}{2} (c+d x)\right )+1}{\sqrt{\cos (c+d x)}}\right )+\frac{80 C \sin \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\cos (c+d x)}}\right )}{10 d (a (\cos (c+d x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.135, size = 341, normalized size = 2.2 \begin{align*} -{\frac{-1+\cos \left ( dx+c \right ) }{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}{a}^{2}} \left ( -10\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-18\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}+2\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}+7\,A\sqrt{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+18\,A\cos \left ( dx+c \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-C\sqrt{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+8\,A \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{5/2}-2\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+2\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 2.35114, size = 522, normalized size = 3.43 \begin{align*} -\frac{\sqrt{2}{\left ({\left (7 \, A - C\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (7 \, A - C\right )} \cos \left (d x + c\right )^{2} +{\left (7 \, A - C\right )} \cos \left (d x + c\right )\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 \,{\left ({\left (5 \, A + C\right )} \cos \left (d x + c\right ) + 4 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \,{\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\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{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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