Optimal. Leaf size=515 \[ \frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d} \]
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Rubi [A] time = 1.59025, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{13}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (\frac{1}{2} (8 A b+11 a B)+\frac{1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac{1}{2} b (A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right )+\frac{1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac{1}{4} b (17 A b+11 a B+99 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right )+\frac{1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac{231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos (c+d x)-\frac{5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{3465}\\ &=\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{693}{32} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right )-\frac{45}{32} \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{10395}\\ &=\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{15} \left (\left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{231} \left (\left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac{1}{15} \left (\left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 7.37236, size = 563, normalized size = 1.09 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{2}{15} \sin (c+d x) \left (28 a^3 A b+54 a^2 b^2 B+36 a^3 b C+7 a^4 B+36 a A b^3+60 a b^3 C+15 b^4 B\right )+\frac{2}{77} \sec ^3(c+d x) \left (66 a^2 A b^2 \sin (c+d x)+9 a^4 A \sin (c+d x)+44 a^3 b B \sin (c+d x)+11 a^4 C \sin (c+d x)\right )+\frac{2}{45} \sec ^2(c+d x) \left (28 a^3 A b \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+36 a^3 b C \sin (c+d x)+7 a^4 B \sin (c+d x)+36 a A b^3 \sin (c+d x)\right )+\frac{2}{231} \sec (c+d x) \left (330 a^2 A b^2 \sin (c+d x)+45 a^4 A \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)+220 a^3 b B \sin (c+d x)+55 a^4 C \sin (c+d x)+308 a b^3 B \sin (c+d x)+77 A b^4 \sin (c+d x)\right )+\frac{2}{9} \sec ^4(c+d x) \left (4 a^3 A b \sin (c+d x)+a^4 B \sin (c+d x)\right )+\frac{2}{11} a^4 A \tan (c+d x) \sec ^4(c+d x)\right )}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (1650 a^2 A b^2+225 a^4 A+2310 a^2 b^2 C+1100 a^3 b B+275 a^4 C+1540 a b^3 B+385 A b^4+1155 b^4 C\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-2156 a^3 A b-4158 a^2 b^2 B-2772 a^3 b C-539 a^4 B-2772 a A b^3-4620 a b^3 C-1155 b^4 B\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{1155 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.52, size = 1550, normalized size = 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \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 ({\left (C b^{4} \cos \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sec \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{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \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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