Optimal. Leaf size=334 \[ \frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
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Rubi [A] time = 1.17617, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3043, 2975, 2980, 2772, 2771} \[ \frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3043
Rule 2975
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{15}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{15}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (5 A+13 B)+\frac{1}{2} a (6 A+13 C) \cos (c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{4} a^2 (136 A+182 B+143 C)+\frac{1}{4} a^2 (96 A+78 B+143 C) \cos (c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac{2 a^2 (136 A+182 B+143 C) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{8} a^3 (2224 A+2522 B+2717 C)+\frac{3}{8} a^3 (560 A+598 B+715 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac{2 a^3 (2224 A+2522 B+2717 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+182 B+143 C) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (a^2 (8368 A+9230 B+10439 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{3003}\\ &=\frac{2 a^3 (8368 A+9230 B+10439 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+182 B+143 C) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (4 a^2 (8368 A+9230 B+10439 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{15015}\\ &=\frac{8 a^3 (8368 A+9230 B+10439 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+182 B+143 C) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac{\left (8 a^2 (8368 A+9230 B+10439 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{45045}\\ &=\frac{16 a^3 (8368 A+9230 B+10439 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt{a+a \cos (c+d x)}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (136 A+182 B+143 C) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac{2 a (5 A+13 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac{13}{2}}(c+d x) \sin (c+d x)}{13 d}\\ \end{align*}
Mathematica [A] time = 1.24411, size = 224, normalized size = 0.67 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{13}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (70 (5552 A+5083 B+4576 C) \cos (c+d x)+14 (30334 A+31850 B+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+138450 B \cos (3 (c+d x))+138450 B \cos (4 (c+d x))+18460 B \cos (5 (c+d x))+18460 B \cos (6 (c+d x))+325910 B+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.238, size = 240, normalized size = 0.7 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 66944\,A \left ( \cos \left ( dx+c \right ) \right ) ^{6}+73840\,B \left ( \cos \left ( dx+c \right ) \right ) ^{6}+83512\,C \left ( \cos \left ( dx+c \right ) \right ) ^{6}+33472\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+36920\,B \left ( \cos \left ( dx+c \right ) \right ) ^{5}+41756\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+25104\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+27690\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+31317\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+20920\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+23075\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+18590\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+18305\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+14560\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5005\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+11970\,A\cos \left ( dx+c \right ) +4095\,B\cos \left ( dx+c \right ) +3465\,A \right ) \cos \left ( dx+c \right ) }{45045\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{15}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.97296, size = 1542, normalized size = 4.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71169, size = 549, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (8 \,{\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \,{\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \,{\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \,{\left (4184 \, A + 4615 \, B + 3718 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \,{\left (523 \, A + 416 \, B + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \,{\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \,{\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )} \sqrt{\cos \left (d x + c\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(-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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