Optimal. Leaf size=236 \[ \frac{128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}+\frac{128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{63 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d} \]
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Rubi [A] time = 0.384936, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2691, 2861, 12, 2669, 3767, 8} \[ \frac{128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}+\frac{128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{63 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2861
Rule 12
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}-\frac{1}{9} \int \sec ^8(c+d x) (a+b \sin (c+d x))^6 \left (-8 a^2+7 b^2-a b \sin (c+d x)\right ) \, dx\\ &=\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{1}{63} \int 48 a \left (a^2-b^2\right ) \sec ^6(c+d x) (a+b \sin (c+d x))^5 \, dx\\ &=\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{1}{21} \left (16 a \left (a^2-b^2\right )\right ) \int \sec ^6(c+d x) (a+b \sin (c+d x))^5 \, dx\\ &=\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac{1}{105} \left (16 a \left (a^2-b^2\right )\right ) \int \left (-4 a^2+4 b^2\right ) \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\\ &=\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{1}{105} \left (64 a \left (a^2-b^2\right )^2\right ) \int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\\ &=\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac{1}{315} \left (64 a \left (a^2-b^2\right )^2\right ) \int \left (-2 a^2+2 b^2\right ) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{1}{315} \left (128 a \left (a^2-b^2\right )^3\right ) \int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac{128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{1}{315} \left (128 a^2 \left (a^2-b^2\right )^3\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}-\frac{\left (128 a^2 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{315 d}\\ &=\frac{128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac{64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac{16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac{\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac{128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d}\\ \end{align*}
Mathematica [A] time = 4.49955, size = 313, normalized size = 1.33 \[ \frac{\cos (c+d x) \left (\frac{a \left (8 (a-b) (1-\sin (c+d x)) \left ((a-b) (1-\sin (c+d x)) \left (2 (a-b) (1-\sin (c+d x)) \left ((a-b) (1-\sin (c+d x)) \left (35 (a+b \sin (c+d x))^4-4 (a-b) (1-\sin (c+d x)) \left ((a+b) (\sin (c+d x)+1) \left (\left (2 a^2+6 a b+7 b^2\right ) \sin ^2(c+d x)+6 \left (a^2+3 a b+b^2\right ) \sin (c+d x)+7 a^2+6 a b+2 b^2\right )+5 (a+b \sin (c+d x))^3\right )\right )+7 (a+b \sin (c+d x))^5\right )+7 (a+b \sin (c+d x))^6\right )+5 (a+b \sin (c+d x))^7\right )+35 (a+b \sin (c+d x))^8\right )}{35 (1-\sin (c+d x))^5 (\sin (c+d x)+1)^4}-\sec ^{10}(c+d x) (a+b \sin (c+d x))^9\right )}{9 d (a-b)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 662, normalized size = 2.8 \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 = 1.00529, size = 425, normalized size = 1.8 \begin{align*} \frac{35 \, b^{8} \tan \left (d x + c\right )^{9} +{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \,{\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 70 \,{\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + 140 \,{\left (7 \, \tan \left (d x + c\right )^{9} + 9 \, \tan \left (d x + c\right )^{7}\right )} a^{2} b^{6} - \frac{280 \,{\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac{56 \,{\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{9}} - \frac{8 \,{\left (105 \, \cos \left (d x + c\right )^{6} - 189 \, \cos \left (d x + c\right )^{4} + 135 \, \cos \left (d x + c\right )^{2} - 35\right )} a b^{7}}{\cos \left (d x + c\right )^{9}} + \frac{280 \, a^{7} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.14994, size = 795, normalized size = 3.37 \begin{align*} -\frac{840 \, a b^{7} \cos \left (d x + c\right )^{6} - 280 \, a^{7} b - 1960 \, a^{5} b^{3} - 1960 \, a^{3} b^{5} - 280 \, a b^{7} - 504 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 360 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} -{\left ({\left (128 \, a^{8} - 448 \, a^{6} b^{2} + 560 \, a^{4} b^{4} - 280 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 35 \, a^{8} + 980 \, a^{6} b^{2} + 2450 \, a^{4} b^{4} + 980 \, a^{2} b^{6} + 35 \, b^{8} + 4 \,{\left (16 \, a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4} - 35 \, a^{2} b^{6} - 35 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \,{\left (8 \, a^{8} - 28 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 350 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 20 \,{\left (2 \, a^{8} - 7 \, a^{6} b^{2} - 175 \, a^{4} b^{4} - 133 \, a^{2} b^{6} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \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 [B] time = 1.19109, size = 1204, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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