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 = 0.38, antiderivative size = 236, normalized size of antiderivative = 1.00, 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 8
Rule 12
Rule 2669
Rule 2691
Rule 2861
Rule 3767
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*}
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Mathematica [A] time = 4.46, 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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fricas [A] time = 0.51, size = 336, normalized size = 1.42 \[ -\frac {840 \, a b^{7} \cos \left (d x + c\right )^{6} - 280 \, a^{7} b - 1960 \, a^{5} b^{3} - 1960 \, a^{3} b^{5} - 280 \, a b^{7} - 504 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 360 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (128 \, a^{8} - 448 \, a^{6} b^{2} + 560 \, a^{4} b^{4} - 280 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 35 \, a^{8} + 980 \, a^{6} b^{2} + 2450 \, a^{4} b^{4} + 980 \, a^{2} b^{6} + 35 \, b^{8} + 4 \, {\left (16 \, a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4} - 35 \, a^{2} b^{6} - 35 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (8 \, a^{8} - 28 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 350 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{8} - 7 \, a^{6} b^{2} - 175 \, a^{4} b^{4} - 133 \, a^{2} b^{6} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.92, size = 892, normalized size = 3.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 662, normalized size = 2.81 \[ \frac {-a^{8} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{9 \cos \left (d x +c \right )^{9}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin ^{6}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{105}\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin ^{8}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{315 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {b^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9 \cos \left (d x +c \right )^{9}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 315, normalized size = 1.33 \[ \frac {35 \, b^{8} \tan \left (d x + c\right )^{9} + {\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 70 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + 140 \, {\left (7 \, \tan \left (d x + c\right )^{9} + 9 \, \tan \left (d x + c\right )^{7}\right )} a^{2} b^{6} - \frac {280 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac {56 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{9}} - \frac {8 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 189 \, \cos \left (d x + c\right )^{4} + 135 \, \cos \left (d x + c\right )^{2} - 35\right )} a b^{7}}{\cos \left (d x + c\right )^{9}} + \frac {280 \, a^{7} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.68, size = 659, normalized size = 2.79 \[ \frac {{\left (a-b\right )}^8}{2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^8}-\frac {{\left (a+b\right )}^8}{9\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^9}-\frac {{\left (a+b\right )}^8}{2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^8}-\frac {{\left (a-b\right )}^8}{9\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9}-\frac {{\left (a+b\right )}^7\,\left (37\,a+21\,b\right )}{28\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^7}-\frac {{\left (a+b\right )}^7\,\left (55\,a+7\,b\right )}{24\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^6}+\frac {{\left (a-b\right )}^5\,\left (187\,a^3+191\,a^2\,b+65\,a\,b^2+5\,b^3\right )}{128\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^2}+\frac {{\left (a-b\right )}^5\,\left (-463\,a^3-67\,a^2\,b+67\,a\,b^2+15\,b^3\right )}{192\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3}+\frac {{\left (a-b\right )}^6\,\left (95\,a^2+18\,a\,b-b^2\right )}{32\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^4}+\frac {{\left (a-b\right )}^6\,\left (-241\,a^2+114\,a\,b+15\,b^2\right )}{80\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5}-\frac {{\left (a-b\right )}^7\,\left (37\,a-21\,b\right )}{28\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7}+\frac {{\left (a-b\right )}^7\,\left (55\,a-7\,b\right )}{24\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^6}+\frac {{\left (a+b\right )}^6\,\left (-95\,a^2+18\,a\,b+b^2\right )}{32\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^4}-\frac {{\left (a+b\right )}^5\,\left (187\,a^3-191\,a^2\,b+65\,a\,b^2-5\,b^3\right )}{128\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^2}+\frac {{\left (a+b\right )}^5\,\left (-463\,a^3+67\,a^2\,b+67\,a\,b^2-15\,b^3\right )}{192\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3}-\frac {{\left (a+b\right )}^6\,\left (241\,a^2+114\,a\,b-15\,b^2\right )}{80\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^5}-\frac {a\,{\left (a+b\right )}^4\,\left (16\,a^3-29\,a^2\,b+20\,a\,b^2-5\,b^3\right )}{16\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}-\frac {a\,{\left (a-b\right )}^4\,\left (16\,a^3+29\,a^2\,b+20\,a\,b^2+5\,b^3\right )}{16\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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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