Optimal. Leaf size=624 \[ \frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{240 d}+\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{960 a d}-\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{1920 a^2 d}-\frac {\left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 2.71, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {\tan (c+d x) \left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{1920 a^2 d}-\frac {\left (-4 a^2 b^2 (809 A+1180 C)-256 a^4 (4 A+5 C)-3560 a^3 b B-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{240 d}+\frac {\tan (c+d x) \sec (c+d x) \left (4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{960 a d}+\frac {(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 3002
Rule 3047
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{5} \int (a+b \cos (c+d x))^{3/2} \left (\frac {5}{2} (A b+2 a B)+(4 a A+5 b B+5 a C) \cos (c+d x)+\frac {1}{2} b (3 A+10 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{20} \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} \left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right )+\frac {1}{2} \left (30 a^2 B+40 b^2 B+a b (59 A+80 C)\right ) \cos (c+d x)+\frac {1}{4} b (39 A b+30 a B+80 b C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{60} \int \frac {\left (\frac {1}{8} \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right )+\frac {1}{4} \left (490 a^2 b B+240 b^3 B+32 a^3 (4 A+5 C)+3 a b^2 (167 A+240 C)\right ) \cos (c+d x)+\frac {3}{8} b \left (170 a b B+16 a^2 (4 A+5 C)+b^2 (93 A+160 C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \frac {\left (\frac {1}{16} \left (1024 a^4 A+1692 a^2 A b^2-45 A b^4+2840 a^3 b B+150 a b^3 B+1280 a^4 C+2640 a^2 b^2 C\right )+\frac {1}{8} a \left (360 a^3 B+1610 a b^2 B+3 b^3 (191 A+320 C)+4 a^2 b (289 A+380 C)\right ) \cos (c+d x)+\frac {1}{16} b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a}\\ &=-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\int \frac {\left (\frac {15}{32} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\frac {1}{16} a b \left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x)+\frac {1}{32} b \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a^2}\\ &=-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {\int \frac {\left (-\frac {15}{32} b \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right )+\frac {1}{32} a b \left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{120 a^2 b}+\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3840 a^2}\\ &=-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{256 a^2}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3840 a}+\frac {\left (\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{3840 a^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {\left (\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{256 a^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{3840 a \sqrt {a+b \cos (c+d x)}}\\ &=\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end {align*}
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Mathematica [C] time = 7.34, size = 930, normalized size = 1.49 \[ \frac {\frac {2 \left (1440 b B a^4+3088 A b^2 a^3+4160 b^2 C a^3+2360 b^3 B a^2+60 A b^4 a\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2880 B a^5+6176 A b a^4+8320 b C a^4+4360 b^2 B a^3-492 A b^3 a^2-240 b^3 C a^2-450 b^4 B a+135 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (45 A b^5-150 a B b^4-1692 a^2 A b^3-2640 a^2 C b^3-2840 a^3 B b^2-1024 a^4 A b-1280 a^4 C b\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right )}}{7680 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{40} \left (10 B \sin (c+d x) a^2+21 A b \sin (c+d x) a\right ) \sec ^4(c+d x)+\frac {1}{5} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac {1}{240} \left (64 A \sin (c+d x) a^2+80 C \sin (c+d x) a^2+170 b B \sin (c+d x) a+93 A b^2 \sin (c+d x)\right ) \sec ^3(c+d x)+\frac {\left (360 B \sin (c+d x) a^3+772 A b \sin (c+d x) a^2+1040 b C \sin (c+d x) a^2+590 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right ) \sec ^2(c+d x)}{960 a}+\frac {\left (1024 A \sin (c+d x) a^4+1280 C \sin (c+d x) a^4+2840 b B \sin (c+d x) a^3+1692 A b^2 \sin (c+d x) a^2+2640 b^2 C \sin (c+d x) a^2+150 b^3 B \sin (c+d x) a-45 A b^4 \sin (c+d x)\right ) \sec (c+d x)}{1920 a^2}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 19.25, size = 5171, normalized size = 8.29 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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