Optimal. Leaf size=404 \[ \frac{4 a b \left (-88 a^4 b^2+125 a^2 b^4+24 a^6-96 b^6\right ) \cos (c+d x)}{105 d}+\frac{b^2 \left (-152 a^4 b^2+174 a^2 b^4+48 a^6-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{105 d}+\frac{2 b \left (8 a^2 b^2+24 a^4-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{2 a b \left (-40 a^2 b^2+24 a^4+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (8 a^2 b^2+24 a^4-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x \]
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Rubi [A] time = 0.820547, antiderivative size = 404, 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} \[ \frac{4 a b \left (-88 a^4 b^2+125 a^2 b^4+24 a^6-96 b^6\right ) \cos (c+d x)}{105 d}+\frac{b^2 \left (-152 a^4 b^2+174 a^2 b^4+48 a^6-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{105 d}+\frac{2 b \left (8 a^2 b^2+24 a^4-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{2 a b \left (-40 a^2 b^2+24 a^4+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (8 a^2 b^2+24 a^4-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2861
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x))^6 \left (-6 a^2+7 b^2+a b \sin (c+d x)\right ) \, dx\\ &=\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}+\frac{1}{35} \int \sec ^4(c+d x) (a+b \sin (c+d x))^5 \left (2 a \left (12 a^2-11 b^2\right )-2 b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{1}{105} \int \sec ^2(c+d x) (a+b \sin (c+d x))^4 \left (-2 \left (24 a^4+8 a^2 b^2-35 b^4\right )+6 a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac{1}{105} \int (a+b \sin (c+d x))^3 \left (24 a b^2 \left (12 a^2-11 b^2\right )-8 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac{1}{420} \int (a+b \sin (c+d x))^2 \left (24 b^2 \left (24 a^4-52 a^2 b^2+35 b^4\right )-24 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac{2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}+\frac{\int (a+b \sin (c+d x)) \left (24 a b^2 \left (24 a^4-76 a^2 b^2+87 b^4\right )-24 b \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x)\right ) \, dx}{1260}\\ &=b^8 x+\frac{4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac{b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{105 d}+\frac{2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac{2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d}\\ \end{align*}
Mathematica [A] time = 1.48119, size = 479, normalized size = 1.19 \[ \frac{\sec ^7(c+d x) \left (23520 a^6 b^2 \sin (c+d x)-4704 a^6 b^2 \sin (3 (c+d x))-1568 a^6 b^2 \sin (5 (c+d x))-224 a^6 b^2 \sin (7 (c+d x))+44100 a^4 b^4 \sin (c+d x)-20580 a^4 b^4 \sin (3 (c+d x))+2940 a^4 b^4 \sin (5 (c+d x))+420 a^4 b^4 \sin (7 (c+d x))+14700 a^2 b^6 \sin (c+d x)-8820 a^2 b^6 \sin (3 (c+d x))+2940 a^2 b^6 \sin (5 (c+d x))-420 a^2 b^6 \sin (7 (c+d x))-37632 a^5 b^3 \cos (2 (c+d x))-12544 a^3 b^5 \cos (2 (c+d x))+15680 a^3 b^5 \cos (4 (c+d x))+16128 a^5 b^3+25536 a^3 b^5+7680 a^7 b+1680 a^8 \sin (c+d x)+1008 a^8 \sin (3 (c+d x))+336 a^8 \sin (5 (c+d x))+48 a^8 \sin (7 (c+d x))-14448 a b^7 \cos (2 (c+d x))-3360 a b^7 \cos (4 (c+d x))-1680 a b^7 \cos (6 (c+d x))-5088 a b^7-1176 b^8 \sin (3 (c+d x))-392 b^8 \sin (5 (c+d x))-176 b^8 \sin (7 (c+d x))+3675 b^8 (c+d x) \cos (c+d x)+2205 b^8 (c+d x) \cos (3 (c+d x))+735 b^8 (c+d x) \cos (5 (c+d x))+105 b^8 (c+d x) \cos (7 (c+d x))\right )}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.14, size = 567, 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 = 1.48954, size = 419, normalized size = 1.04 \begin{align*} \frac{420 \, a^{2} b^{6} \tan \left (d x + c\right )^{7} + 3 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \,{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 210 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} +{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} b^{8} - \frac{168 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac{56 \,{\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{7}} - \frac{24 \,{\left (35 \, \cos \left (d x + c\right )^{6} - 35 \, \cos \left (d x + c\right )^{4} + 21 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{7}}{\cos \left (d x + c\right )^{7}} + \frac{120 \, a^{7} b}{\cos \left (d x + c\right )^{7}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.90773, size = 732, normalized size = 1.81 \begin{align*} \frac{105 \, b^{8} d x \cos \left (d x + c\right )^{7} - 840 \, a b^{7} \cos \left (d x + c\right )^{6} + 120 \, a^{7} b + 840 \, a^{5} b^{3} + 840 \, a^{3} b^{5} + 120 \, a b^{7} + 280 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 168 \,{\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 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} + 4 \,{\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} - 105 \, a^{2} b^{6} - 44 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \,{\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} + 630 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (3 \, a^{8} - 14 \, a^{6} b^{2} - 280 \, a^{4} b^{4} - 210 \, a^{2} b^{6} - 11 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \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 = 1.19368, size = 980, normalized size = 2.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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