Optimal. Leaf size=381 \[ \frac{2 a b \left (-48 a^4 b^2+163 a^2 b^4+8 a^6+192 b^6\right ) \cos (c+d x)}{15 d}+\frac{4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac{b \left (-16 a^2 b^2+8 a^4+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac{a b \left (-32 a^2 b^2+8 a^4+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac{b^2 \left (-88 a^4 b^2+282 a^2 b^4+16 a^6+105 b^6\right ) \sin (c+d x) \cos (c+d x)}{30 d}-\frac{\sec ^3(c+d x) \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{15 d}-\frac{4 \sec (c+d x) \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^5}{15 d}-\frac{7}{2} b^6 x \left (8 a^2+b^2\right )+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d} \]
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Rubi [A] time = 0.723576, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2691, 2861, 2753, 2734} \[ \frac{2 a b \left (-48 a^4 b^2+163 a^2 b^4+8 a^6+192 b^6\right ) \cos (c+d x)}{15 d}+\frac{4 a b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac{b \left (-16 a^2 b^2+8 a^4+35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac{a b \left (-32 a^2 b^2+8 a^4+87 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac{b^2 \left (-88 a^4 b^2+282 a^2 b^4+16 a^6+105 b^6\right ) \sin (c+d x) \cos (c+d x)}{30 d}-\frac{\sec ^3(c+d x) \left (3 a b-\left (4 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{15 d}-\frac{4 \sec (c+d x) \left (b \left (4 a^2-7 b^2\right )-a \left (2 a^2+b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^5}{15 d}-\frac{7}{2} b^6 x \left (8 a^2+b^2\right )+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2861
Rule 2753
Rule 2734
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*}
Mathematica [A] time = 1.29166, size = 472, normalized size = 1.24 \[ \frac{\sec ^5(c+d x) \left (8960 a^6 b^2 \sin (c+d x)-2240 a^6 b^2 \sin (3 (c+d x))-448 a^6 b^2 \sin (5 (c+d x))+16800 a^4 b^4 \sin (c+d x)-8400 a^4 b^4 \sin (3 (c+d x))+1680 a^4 b^4 \sin (5 (c+d x))+11200 a^2 b^6 \sin (c+d x)+5600 a^2 b^6 \sin (3 (c+d x))+5152 a^2 b^6 \sin (5 (c+d x))-17920 a^5 b^3 \cos (2 (c+d x))+17920 a^3 b^5 \cos (2 (c+d x))+13440 a^3 b^5 \cos (4 (c+d x))-33600 a^2 b^6 (c+d x) \cos (c+d x)-16800 a^2 b^6 (c+d x) \cos (3 (c+d x))-3360 a^2 b^6 (c+d x) \cos (5 (c+d x))+3584 a^5 b^3+25984 a^3 b^5+3072 a^7 b+640 a^8 \sin (c+d x)+320 a^8 \sin (3 (c+d x))+64 a^8 \sin (5 (c+d x))+22560 a b^7 \cos (2 (c+d x))+8640 a b^7 \cos (4 (c+d x))+480 a b^7 \cos (6 (c+d x))+17472 a b^7+875 b^8 \sin (c+d x)+1015 b^8 \sin (3 (c+d x))+539 b^8 \sin (5 (c+d x))+15 b^8 \sin (7 (c+d x))-4200 b^8 (c+d x) \cos (c+d x)-2100 b^8 (c+d x) \cos (3 (c+d x))-420 b^8 (c+d x) \cos (5 (c+d x))\right )}{1920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 544, normalized size = 1.4 \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 [A] time = 1.46795, size = 425, normalized size = 1.12 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.01358, size = 660, normalized size = 1.73 \begin{align*} \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{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 [A] time = 1.19863, size = 895, normalized size = 2.35 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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