Optimal. Leaf size=192 \[ \frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{63 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.22, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2691, 2861, 2669, 3767} \[ \frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{63 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2691
Rule 2861
Rule 3767
Rubi steps
\begin {align*} \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}-\frac {1}{9} \int \sec ^8(c+d x) (a+b \sin (c+d x)) \left (-8 a^2+2 b^2-6 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))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{63} \int \sec ^6(c+d x) \left (6 a \left (8 a^2-3 b^2\right )+10 b \left (4 a^2-b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{21} \left (2 a \left (8 a^2-3 b^2\right )\right ) \int \sec ^6(c+d x) \, dx\\ &=\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}-\frac {\left (2 a \left (8 a^2-3 b^2\right )\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{21 d}\\ &=\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 299, normalized size = 1.56 \[ \frac {\sec ^9(c+d x) \left (2064384 a^3 \sin (c+d x)+1376256 a^3 \sin (3 (c+d x))+589824 a^3 \sin (5 (c+d x))+147456 a^3 \sin (7 (c+d x))+16384 a^3 \sin (9 (c+d x))+3150 b \left (23 b^2-147 a^2\right ) \cos (c+d x)-308700 a^2 b \cos (3 (c+d x))-132300 a^2 b \cos (5 (c+d x))-33075 a^2 b \cos (7 (c+d x))-3675 a^2 b \cos (9 (c+d x))+3440640 a^2 b+3096576 a b^2 \sin (c+d x)-516096 a b^2 \sin (3 (c+d x))-221184 a b^2 \sin (5 (c+d x))-55296 a b^2 \sin (7 (c+d x))-6144 a b^2 \sin (9 (c+d x))-737280 b^3 \cos (2 (c+d x))+48300 b^3 \cos (3 (c+d x))+20700 b^3 \cos (5 (c+d x))+5175 b^3 \cos (7 (c+d x))+575 b^3 \cos (9 (c+d x))+409600 b^3\right )}{10321920 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 146, normalized size = 0.76 \[ -\frac {45 \, b^{3} \cos \left (d x + c\right )^{2} - 105 \, a^{2} b - 35 \, b^{3} - {\left (16 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} + 8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 35 \, a^{3} + 105 \, a b^{2} + 5 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.60, size = 473, normalized size = 2.46 \[ -\frac {2 \, {\left (315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 945 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 1260 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 630 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 4788 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1512 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 8820 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1050 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 5112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 8532 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3150 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 10658 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4272 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13230 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1890 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 5112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8532 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1890 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4788 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1512 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3780 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 270 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 90 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, a^{2} b - 10 \, b^{3}\right )}}{315 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{9} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 265, normalized size = 1.38 \[ \frac {-a^{3} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{3 \cos \left (d x +c \right )^{9}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 145, normalized size = 0.76 \[ \frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac {105 \, a^{2} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.13, size = 275, normalized size = 1.43 \[ \frac {b^3}{9\,d\,{\cos \left (c+d\,x\right )}^9}-\frac {b^3}{7\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a^2\,b}{3\,d\,{\cos \left (c+d\,x\right )}^9}+\frac {128\,a^3\,\sin \left (c+d\,x\right )}{315\,d\,\cos \left (c+d\,x\right )}+\frac {64\,a^3\,\sin \left (c+d\,x\right )}{315\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {16\,a^3\,\sin \left (c+d\,x\right )}{105\,d\,{\cos \left (c+d\,x\right )}^5}+\frac {8\,a^3\,\sin \left (c+d\,x\right )}{63\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a^3\,\sin \left (c+d\,x\right )}{9\,d\,{\cos \left (c+d\,x\right )}^9}-\frac {16\,a\,b^2\,\sin \left (c+d\,x\right )}{105\,d\,\cos \left (c+d\,x\right )}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{105\,d\,{\cos \left (c+d\,x\right )}^3}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{35\,d\,{\cos \left (c+d\,x\right )}^5}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{21\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a\,b^2\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^9} \]
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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