Optimal. Leaf size=227 \[ \frac{(15 A-39 B+95 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.686808, antiderivative size = 227, 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, 2977, 2968, 3023, 2751, 2649, 206} \[ \frac{(15 A-39 B+95 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(19 A-75 B+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2977
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))^{5/2}} \, dx &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{\int \frac{\cos ^2(c+d x) \left (a (A+3 B-3 C)+\frac{1}{2} a (3 A-3 B+11 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos (c+d x) \left (-a^2 (A-9 B+17 C)+\frac{1}{4} a^2 (15 A-39 B+95 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{-a^2 (A-9 B+17 C) \cos (c+d x)+\frac{1}{4} a^2 (15 A-39 B+95 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(15 A-39 B+95 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac{\int \frac{\frac{1}{8} a^3 (15 A-39 B+95 C)-\frac{1}{4} a^3 (21 A-93 B+197 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{12 a^5}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(15 A-39 B+95 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}+\frac{(19 A-75 B+163 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(15 A-39 B+95 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}-\frac{(19 A-75 B+163 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 )}{16 a^2 d}\\ &=\frac{(19 A-75 B+163 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(A-9 B+17 C) \cos ^2(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{(15 A-39 B+95 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{48 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.16389, size = 126, normalized size = 0.56 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) ((-39 A+255 B-479 C) \cos (c+d x)-27 A+16 (3 B-5 C) \cos (2 (c+d x))+195 B+8 C \cos (3 (c+d x))-379 C)+6 (19 A-75 B+163 C) \cos ^3\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{48 a d (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.157, size = 512, 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 = 2.0934, size = 743, normalized size = 3.27 \begin{align*} \frac{3 \, \sqrt{2}{\left ({\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (19 \, A - 75 \, B + 163 \, C\right )} \cos \left (d x + c\right ) + 19 \, A - 75 \, B + 163 \, 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 (32 \, C \cos \left (d x + c\right )^{3} + 32 \,{\left (3 \, B - 5 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (39 \, A - 255 \, B + 503 \, C\right )} \cos \left (d x + c\right ) - 27 \, A + 147 \, B - 299 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{192 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 = 3.23447, size = 311, normalized size = 1.37 \begin{align*} \frac{\frac{{\left ({\left (3 \,{\left (\frac{2 \, \sqrt{2}{\left (A a^{5} - B a^{5} + C a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{6}} - \frac{\sqrt{2}{\left (7 \, A a^{5} - 15 \, B a^{5} + 23 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{4 \, \sqrt{2}{\left (15 \, A a^{5} - 75 \, B a^{5} + 167 \, C a^{5}\right )}}{a^{6}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \frac{3 \, \sqrt{2}{\left (11 \, A a^{5} - 83 \, B a^{5} + 155 \, C a^{5}\right )}}{a^{6}}\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{3}{2}}} - \frac{3 \, \sqrt{2}{\left (19 \, A - 75 \, B + 163 \, 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{5}{2}}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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