Optimal. Leaf size=229 \[ -\frac{(15 A-35 B+39 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.669542, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3041, 2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{(15 A-35 B+39 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos ^2(c+d x) \left (-a (A-3 B+3 C)+\frac{1}{2} a (5 A-5 B+9 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos (c+d x) \left (a^2 (5 A-5 B+9 C)-\frac{1}{4} a^2 (15 A-35 B+39 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{5 a^3}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{a^2 (5 A-5 B+9 C) \cos (c+d x)-\frac{1}{4} a^2 (15 A-35 B+39 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{5 a^3}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{(15 A-35 B+39 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{30 a^2 d}+\frac{2 \int \frac{-\frac{1}{8} a^3 (15 A-35 B+39 C)+\frac{1}{4} a^3 (45 A-65 B+93 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a+a \cos (c+d x)}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{(15 A-35 B+39 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{30 a^2 d}-\frac{(7 A-11 B+15 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a+a \cos (c+d x)}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{(15 A-35 B+39 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{30 a^2 d}+\frac{(7 A-11 B+15 C) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{2 a d}\\ &=-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a+a \cos (c+d x)}}+\frac{(5 A-5 B+9 C) \cos ^2(c+d x) \sin (c+d x)}{10 a d \sqrt{a+a \cos (c+d x)}}-\frac{(15 A-35 B+39 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{30 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.722273, size = 153, normalized size = 0.67 \[ \frac{15 (7 A-11 B+15 C) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) (3 (20 A-20 B+39 C) \cos (c+d x)+75 A+2 (5 B-3 C) \cos (2 (c+d x))-85 B+3 C \cos (3 (c+d x))+141 C)}{15 d \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.159, size = 533, normalized size = 2.3 \begin{align*} \text{result too large to display} \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 = 1.98861, size = 641, normalized size = 2.8 \begin{align*} \frac{15 \, \sqrt{2}{\left ({\left (7 \, A - 11 \, B + 15 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (7 \, A - 11 \, B + 15 \, C\right )} \cos \left (d x + c\right ) + 7 \, A - 11 \, B + 15 \, C\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \,{\left (12 \, C \cos \left (d x + c\right )^{3} + 4 \,{\left (5 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{2} + 12 \,{\left (5 \, A - 5 \, B + 9 \, C\right )} \cos \left (d x + c\right ) + 75 \, A - 95 \, B + 147 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{120 \,{\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 [A] time = 2.59542, size = 308, normalized size = 1.34 \begin{align*} \frac{\frac{15 \, \sqrt{2}{\left (7 \, A - 11 \, B + 15 \, C\right )} \log \left ({\left | -\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac{3}{2}}} + \frac{{\left ({\left ({\left (\frac{15 \, \sqrt{2}{\left (A a^{3} - B a^{3} + C a^{3}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{2}} + \frac{\sqrt{2}{\left (165 \, A a^{3} - 245 \, B a^{3} + 381 \, C a^{3}\right )}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{5 \, \sqrt{2}{\left (57 \, A a^{3} - 73 \, B a^{3} + 105 \, C a^{3}\right )}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{15 \, \sqrt{2}{\left (9 \, A a^{3} - 9 \, B a^{3} + 17 \, C a^{3}\right )}}{a^{2}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{5}{2}}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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