Optimal. Leaf size=285 \[ \frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.93092, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {4221, 3042, 2983, 2982, 2782, 205, 2774, 216} \[ \frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3042
Rule 2983
Rule 2982
Rule 2782
Rule 205
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx\\ &=-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (-\frac{1}{2} a (A+5 C)+2 a (A+2 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (3 a^2 (A+2 C)-a^2 (2 A+7 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{2} a^3 (2 A+7 C)+\frac{1}{2} a^3 (8 A+19 C) \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{4 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left ((5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{4 a}+\frac{\left ((8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{8 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left ((5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{\left ((8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 )}{4 a^2 d}\\ &=\frac{(8 A+19 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{4 a^{3/2} d}-\frac{(5 A+13 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{\cos (c+d x)} \sqrt{\sec (c+d x)}}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.19786, size = 385, normalized size = 1.35 \[ -\frac{i e^{\frac{1}{2} i (c+d x)} \cos \left (\frac{1}{2} (c+d x)\right ) \left (\frac{\sqrt{2} e^{-2 i (c+d x)} \left (e^{i (c+d x)}-e^{2 i (c+d x)}+e^{3 i (c+d x)}-1\right ) \left (C \left (-3 e^{i (c+d x)}-12 e^{2 i (c+d x)}-3 e^{3 i (c+d x)}+e^{4 i (c+d x)}+1\right )-4 A e^{2 i (c+d x)}\right )}{\sqrt{1+e^{2 i (c+d x)}}}+\sqrt{2} (8 A+19 C) \left (1+e^{i (c+d x)}\right )^2 \sinh ^{-1}\left (e^{i (c+d x)}\right )+4 (5 A+13 C) \left (1+e^{i (c+d x)}\right )^2 \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )-\sqrt{2} (8 A+19 C) \left (1+e^{i (c+d x)}\right )^2 \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{16 d \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{3/2} \left (1+e^{2 i (c+d x)}\right )^{3/2} (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.199, size = 404, normalized size = 1.4 \begin{align*} -{\frac{\sqrt{2}\cos \left ( dx+c \right ) \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3}}{8\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( -2\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+5\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+8\,A\sqrt{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 ) \sin \left ( dx+c \right ) +2\,A\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) +19\,C\sqrt{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 ) \sin \left ( dx+c \right ) +4\,C\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) +10\,A\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) -2\,A\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+26\,C\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) -7\,C\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}} \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{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \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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Fricas [A] time = 75.4157, size = 701, normalized size = 2.46 \begin{align*} \frac{\sqrt{2}{\left ({\left (5 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (5 \, A + 13 \, C\right )} \cos \left (d x + c\right ) + 5 \, A + 13 \, C\right )} \sqrt{a} \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 ) -{\left ({\left (8 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, A + 19 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 19 \, 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{{\left (2 \, C \cos \left (d x + c\right )^{3} - 3 \, C \cos \left (d x + c\right )^{2} -{\left (2 \, A + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{4 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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{C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \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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