Optimal. Leaf size=277 \[ \frac{(245 A-273 B+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A-15 B+19 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.872587, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(245 A-273 B+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A-15 B+19 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]
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 ^3(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 ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos ^3(c+d x) \left (-2 a (A-2 B+2 C)+\frac{1}{2} a (7 A-7 B+11 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos ^2(c+d x) \left (\frac{3}{2} a^2 (7 A-7 B+11 C)-\frac{1}{4} a^2 (35 A-63 B+67 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{7 a^3}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\cos (c+d x) \left (-\frac{1}{2} a^3 (35 A-63 B+67 C)+\frac{1}{8} a^3 (245 A-273 B+397 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{-\frac{1}{2} a^3 (35 A-63 B+67 C) \cos (c+d x)+\frac{1}{8} a^3 (245 A-273 B+397 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A-273 B+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{4 \int \frac{\frac{1}{16} a^4 (245 A-273 B+397 C)-\frac{1}{8} a^4 (455 A-651 B+799 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{105 a^5}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A-273 B+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{(11 A-15 B+19 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A-273 B+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}-\frac{(11 A-15 B+19 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{(11 A-15 B+19 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 ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A-273 B+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.50264, size = 180, normalized size = 0.65 \[ \frac{\frac{1}{2} \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) (6 (140 A-273 B+277 C) \cos (c+d x)-4 (35 A-21 B+64 C) \cos (2 (c+d x))+1190 A-42 B \cos (3 (c+d x))-1974 B+18 C \cos (3 (c+d x))-15 C \cos (4 (c+d x))+2161 C)-105 (11 A-15 B+19 C) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{105 d \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.16, size = 577, normalized size = 2.1 \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 = 2.00992, size = 709, normalized size = 2.56 \begin{align*} \frac{105 \, \sqrt{2}{\left ({\left (11 \, A - 15 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (11 \, A - 15 \, B + 19 \, C\right )} \cos \left (d x + c\right ) + 11 \, A - 15 \, B + 19 \, 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 (60 \, C \cos \left (d x + c\right )^{4} + 12 \,{\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} + 28 \,{\left (5 \, A - 3 \, B + 7 \, C\right )} \cos \left (d x + c\right )^{2} - 12 \,{\left (35 \, A - 63 \, B + 67 \, C\right )} \cos \left (d x + c\right ) - 665 \, A + 1029 \, B - 1201 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{840 \,{\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 = 2.83865, size = 412, normalized size = 1.49 \begin{align*} -\frac{\frac{105 \,{\left (11 \, \sqrt{2} A - 15 \, \sqrt{2} B + 19 \, \sqrt{2} 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 ({\left (\frac{105 \,{\left (\sqrt{2} A a^{5} - \sqrt{2} B a^{5} + \sqrt{2} C a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3}} + \frac{4 \,{\left (455 \, \sqrt{2} A a^{5} - 693 \, \sqrt{2} B a^{5} + 877 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{14 \,{\left (305 \, \sqrt{2} A a^{5} - 453 \, \sqrt{2} B a^{5} + 517 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{140 \,{\left (25 \, \sqrt{2} A a^{5} - 39 \, \sqrt{2} B a^{5} + 47 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{105 \,{\left (9 \, \sqrt{2} A a^{5} - 17 \, \sqrt{2} B a^{5} + 17 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\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{7}{2}}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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