Integrand size = 21, antiderivative size = 165 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d} \]
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Time = 0.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2770, 2940, 2748, 3852} \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (\left (3 a^2-b^2\right ) \sin (c+d x)+2 a b\right )}{35 d}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d} \]
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Rule 2748
Rule 2770
Rule 2940
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x)) \left (-6 a^2+2 b^2-4 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))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \int \sec ^4(c+d x) \left (12 a \left (2 a^2-b^2\right )+6 b \left (3 a^2-b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {1}{35} \left (12 a \left (2 a^2-b^2\right )\right ) \int \sec ^4(c+d x) \, dx \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {\left (12 a \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sec ^3(c+d x)}{35 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d}+\frac {2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left (2 a b+\left (3 a^2-b^2\right ) \sin (c+d x)\right )}{35 d}+\frac {12 a \left (2 a^2-b^2\right ) \tan (c+d x)}{35 d}+\frac {4 a \left (2 a^2-b^2\right ) \tan ^3(c+d x)}{35 d} \\ \end{align*}
Time = 1.73 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.48 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\sec ^7(c+d x) \left (15360 a^2 b+1536 b^3+35 b \left (-75 a^2+17 b^2\right ) \cos (c+d x)-3584 b^3 \cos (2 (c+d x))-1575 a^2 b \cos (3 (c+d x))+357 b^3 \cos (3 (c+d x))-525 a^2 b \cos (5 (c+d x))+119 b^3 \cos (5 (c+d x))-75 a^2 b \cos (7 (c+d x))+17 b^3 \cos (7 (c+d x))+8960 a^3 \sin (c+d x)+13440 a b^2 \sin (c+d x)+5376 a^3 \sin (3 (c+d x))-2688 a b^2 \sin (3 (c+d x))+1792 a^3 \sin (5 (c+d x))-896 a b^2 \sin (5 (c+d x))+256 a^3 \sin (7 (c+d x))-128 a b^2 \sin (7 (c+d x))\right )}{35840 d} \]
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Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {16 \left (70 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+14 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-70 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-35 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-12 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-42 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+21 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+14 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-14 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+7 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{3}+i a \,b^{2}\right )}{35 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{7}}\) | \(199\) |
derivativedivides | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(219\) |
default | \(\frac {-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )}{d}\) | \(219\) |
parallelrisch | \(-\frac {2 \left (35 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+105 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -70 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+140 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+70 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+301 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+112 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+525 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +70 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-212 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+456 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+140 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+301 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+112 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+315 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +28 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-70 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+140 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+14 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+35 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15 a^{2} b -2 b^{3}\right )}{35 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(359\) |
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.75 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {7 \, b^{3} \cos \left (d x + c\right )^{2} - 15 \, a^{2} b - 5 \, b^{3} - {\left (8 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, a^{3} + 15 \, a b^{2} + 3 \, {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d \cos \left (d x + c\right )^{7}} \]
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Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.75 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{3} + {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac {15 \, a^{2} b}{\cos \left (d x + c\right )^{7}}}{35 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (155) = 310\).
Time = 0.35 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.17 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 140 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 301 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 112 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 70 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 212 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 456 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 140 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 301 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 112 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 315 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 28 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 140 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a^{2} b - 2 \, b^{3}\right )}}{35 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7} d} \]
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Time = 5.02 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.92 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {8\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {4\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )+{\cos \left (c+d\,x\right )}^6\,\left (\frac {16\,a^3\,\sin \left (c+d\,x\right )}{35}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{35}\right )-{\cos \left (c+d\,x\right )}^2\,\left (-\frac {6\,\sin \left (c+d\,x\right )\,a^3}{35}+\frac {3\,\sin \left (c+d\,x\right )\,a\,b^2}{35}+\frac {b^3}{5}\right )+\frac {3\,a^2\,b}{7}+\frac {a^3\,\sin \left (c+d\,x\right )}{7}+\frac {b^3}{7}+\frac {3\,a\,b^2\,\sin \left (c+d\,x\right )}{7}}{d\,{\cos \left (c+d\,x\right )}^7} \]
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