Optimal. Leaf size=381 \[ -\frac {7}{2} b^6 \left (8 a^2+b^2\right ) x+\frac {2 a b \left (8 a^6-48 a^4 b^2+163 a^2 b^4+192 b^6\right ) \cos (c+d x)}{15 d}+\frac {b^2 \left (16 a^6-88 a^4 b^2+282 a^2 b^4+105 b^6\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {a b \left (8 a^4-32 a^2 b^2+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac {b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d} \]
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Rubi [A]
time = 0.48, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2770, 2940,
2832, 2813} \begin {gather*} \frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}-\frac {\sec ^3(c+d x) \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{15 d}-\frac {4 \sec (c+d x) \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^5}{15 d}-\frac {7}{2} b^6 x \left (8 a^2+b^2\right )+\frac {b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac {a b \left (8 a^4-32 a^2 b^2+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac {2 a b \left (8 a^6-48 a^4 b^2+163 a^2 b^4+192 b^6\right ) \cos (c+d x)}{15 d}+\frac {b^2 \left (16 a^6-88 a^4 b^2+282 a^2 b^4+105 b^6\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2770
Rule 2813
Rule 2832
Rule 2940
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {1}{5} \int \sec ^4(c+d x) (a+b \sin (c+d x))^6 \left (-4 a^2+7 b^2+3 a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}+\frac {1}{15} \int \sec ^2(c+d x) (a+b \sin (c+d x))^5 \left (4 a \left (2 a^2+b^2\right )-4 b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {1}{15} \int (a+b \sin (c+d x))^4 \left (-20 b^2 \left (4 a^2-7 b^2\right )+20 a b \left (2 a^2+b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {1}{75} \int (a+b \sin (c+d x))^3 \left (-60 a b^2 \left (4 a^2-13 b^2\right )+20 b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {1}{300} \int (a+b \sin (c+d x))^2 \left (-60 b^2 \left (8 a^4-36 a^2 b^2-35 b^4\right )+60 a b \left (8 a^4-32 a^2 b^2+87 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {a b \left (8 a^4-32 a^2 b^2+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac {b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {1}{900} \int (a+b \sin (c+d x)) \left (-60 a b^2 \left (8 a^4-44 a^2 b^2-279 b^4\right )+60 b \left (16 a^6-88 a^4 b^2+282 a^2 b^4+105 b^6\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {7}{2} b^6 \left (8 a^2+b^2\right ) x+\frac {2 a b \left (8 a^6-48 a^4 b^2+163 a^2 b^4+192 b^6\right ) \cos (c+d x)}{15 d}+\frac {b^2 \left (16 a^6-88 a^4 b^2+282 a^2 b^4+105 b^6\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {a b \left (8 a^4-32 a^2 b^2+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac {b \left (8 a^4-16 a^2 b^2+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac {4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{5 d}-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^6 \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 d}-\frac {4 \sec (c+d x) (a+b \sin (c+d x))^5 \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{15 d}\\ \end {align*}
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Mathematica [A]
time = 1.31, size = 472, normalized size = 1.24 \begin {gather*} \frac {\sec ^5(c+d x) \left (3072 a^7 b+3584 a^5 b^3+25984 a^3 b^5+17472 a b^7-33600 a^2 b^6 (c+d x) \cos (c+d x)-4200 b^8 (c+d x) \cos (c+d x)-17920 a^5 b^3 \cos (2 (c+d x))+17920 a^3 b^5 \cos (2 (c+d x))+22560 a b^7 \cos (2 (c+d x))-16800 a^2 b^6 (c+d x) \cos (3 (c+d x))-2100 b^8 (c+d x) \cos (3 (c+d x))+13440 a^3 b^5 \cos (4 (c+d x))+8640 a b^7 \cos (4 (c+d x))-3360 a^2 b^6 (c+d x) \cos (5 (c+d x))-420 b^8 (c+d x) \cos (5 (c+d x))+480 a b^7 \cos (6 (c+d x))+640 a^8 \sin (c+d x)+8960 a^6 b^2 \sin (c+d x)+16800 a^4 b^4 \sin (c+d x)+11200 a^2 b^6 \sin (c+d x)+875 b^8 \sin (c+d x)+320 a^8 \sin (3 (c+d x))-2240 a^6 b^2 \sin (3 (c+d x))-8400 a^4 b^4 \sin (3 (c+d x))+5600 a^2 b^6 \sin (3 (c+d x))+1015 b^8 \sin (3 (c+d x))+64 a^8 \sin (5 (c+d x))-448 a^6 b^2 \sin (5 (c+d x))+1680 a^4 b^4 \sin (5 (c+d x))+5152 a^2 b^6 \sin (5 (c+d x))+539 b^8 \sin (5 (c+d x))+15 b^8 \sin (7 (c+d x))\right )}{1920 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 544, normalized size = 1.43 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 315, normalized size = 0.83 \begin {gather*} \frac {420 \, a^{4} b^{4} \tan \left (d x + c\right )^{5} + 2 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{8} + 56 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 56 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} + {\left (6 \, \tan \left (d x + c\right )^{5} - 20 \, \tan \left (d x + c\right )^{3} - 105 \, d x - 105 \, c + \frac {15 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} + 90 \, \tan \left (d x + c\right )\right )} b^{8} + 48 \, a b^{7} {\left (\frac {15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1}{\cos \left (d x + c\right )^{5}} + 5 \, \cos \left (d x + c\right )\right )} - \frac {112 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{5}} + \frac {112 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} + 3\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{5}} + \frac {48 \, a^{7} b}{\cos \left (d x + c\right )^{5}}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 281, normalized size = 0.74 \begin {gather*} \frac {240 \, a b^{7} \cos \left (d x + c\right )^{6} + 48 \, a^{7} b + 336 \, a^{5} b^{3} + 336 \, a^{3} b^{5} + 48 \, a b^{7} - 105 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} d x \cos \left (d x + c\right )^{5} + 240 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 80 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, b^{8} \cos \left (d x + c\right )^{6} + 6 \, a^{8} + 168 \, a^{6} b^{2} + 420 \, a^{4} b^{4} + 168 \, a^{2} b^{6} + 6 \, b^{8} + 4 \, {\left (4 \, a^{8} - 28 \, a^{6} b^{2} + 105 \, a^{4} b^{4} + 322 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (a^{8} - 7 \, a^{6} b^{2} - 105 \, a^{4} b^{4} - 77 \, a^{2} b^{6} - 4 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{30 \, d \cos \left (d x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.86, size = 663, normalized size = 1.74 \begin {gather*} -\frac {105 \, {\left (8 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )} + \frac {30 \, {\left (b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 16 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {4 \, {\left (15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 20 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2240 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 220 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1680 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 720 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 58 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 224 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3360 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4984 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 398 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4480 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1920 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 20 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 220 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2240 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1200 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 420 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{7} b - 112 \, a^{5} b^{3} + 448 \, a^{3} b^{5} + 264 \, a b^{7}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.60, size = 665, normalized size = 1.75 \begin {gather*} -\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8+56\,a^2\,b^6+7\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (48\,a^7\,b+\frac {1568\,a^5\,b^3}{3}+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (32\,a^7\,b+224\,a^5\,b^3\right )+\frac {256\,a\,b^7}{5}+\frac {16\,a^7\,b}{5}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8+56\,a^2\,b^6+7\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (64\,a^7\,b+448\,a^5\,b^3+896\,a^3\,b^5+256\,a\,b^7\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-\frac {32\,a^7\,b}{5}-\frac {224\,a^5\,b^3}{5}+\frac {896\,a^3\,b^5}{5}+\frac {768\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {176\,a^7\,b}{5}+\frac {3136\,a^5\,b^3}{15}+\frac {896\,a^3\,b^5}{15}+\frac {256\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {22\,a^8}{5}+\frac {896\,a^6\,b^2}{5}+448\,a^4\,b^4+\frac {616\,a^2\,b^6}{5}+\frac {77\,b^8}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {22\,a^8}{5}+\frac {896\,a^6\,b^2}{5}+448\,a^4\,b^4+\frac {616\,a^2\,b^6}{5}+\frac {77\,b^8}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {152\,a^8}{15}+\frac {3136\,a^6\,b^2}{15}+896\,a^4\,b^4+\frac {10976\,a^2\,b^6}{15}+\frac {412\,b^8}{15}\right )+\frac {896\,a^3\,b^5}{15}-\frac {224\,a^5\,b^3}{15}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {4\,a^8}{3}+\frac {224\,a^6\,b^2}{3}-\frac {560\,a^2\,b^6}{3}-\frac {70\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {4\,a^8}{3}+\frac {224\,a^6\,b^2}{3}-\frac {560\,a^2\,b^6}{3}-\frac {70\,b^8}{3}\right )+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {7\,b^6\,\mathrm {atan}\left (\frac {7\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^2+b^2\right )}{56\,a^2\,b^6+7\,b^8}\right )\,\left (8\,a^2+b^2\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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