Optimal. Leaf size=622 \[ \frac{2 \left (1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left (15 a^2 b^2 (19 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^4 d}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]
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Rubi [A] time = 2.6248, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2989, 3047, 3055, 2998, 2816, 2994} \[ \frac{2 \left (1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left (15 a^2 b^2 (19 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3465 a^4 d}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2989
Rule 3047
Rule 3055
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2}{11} \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{2} a (14 A b+11 a B)+\frac{1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \cos (c+d x)+\frac{1}{2} b (6 a A+11 b B) \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{4}{99} \int \frac{\frac{1}{4} a \left (81 a^2 A+113 A b^2+209 a b B\right )+\frac{1}{4} \left (233 a^2 A b+99 A b^3+77 a^3 B+297 a b^2 B\right ) \cos (c+d x)+\frac{3}{4} b \left (46 a A b+22 a^2 B+33 b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{8 \int \frac{\frac{1}{8} a \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right )+\frac{1}{8} a \left (405 a^3 A+1531 a A b^2+1507 a^2 b B+693 b^3 B\right ) \cos (c+d x)+\frac{1}{2} a b \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{693 a}\\ &=\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 \int \frac{\frac{3}{16} a \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right )+\frac{1}{16} a^2 \left (5055 a^2 A b+2305 A b^3+1617 a^3 B+6655 a b^2 B\right ) \cos (c+d x)+\frac{1}{8} a b \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3465 a^2}\\ &=\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{32 \int \frac{\frac{3}{32} a \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right )+\frac{3}{32} a^2 \left (675 a^4 A+3315 a^2 A b^2+10 A b^4+2871 a^3 b B+1705 a b^3 B\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{10395 a^3}\\ &=\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{\left ((a-b) \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{3465 a^2}+\frac{\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3465 a^2}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d}+\frac{2 a (14 A b+11 a B) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}\\ \end{align*}
Mathematica [C] time = 6.80217, size = 1640, normalized size = 2.64 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.94, size = 5373, normalized size = 8.6 \begin{align*} \text{output too large to display} \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{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B b^{2} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac{13}{2}}}, x\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{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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