Optimal. Leaf size=349 \[ \frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{80 d}-\frac {7}{16} b^2 x \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right )+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d} \]
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Rubi [A] time = 0.56, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2691, 2753, 2734} \[ \frac {a b \left (1664 a^4 b^2+2789 a^2 b^4+40 a^6+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (992 a^2 b^2+120 a^4+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (624 a^2 b^2+40 a^4+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b^2 \left (2248 a^4 b^2+2502 a^2 b^4+80 a^6+175 b^6\right ) \sin (c+d x) \cos (c+d x)}{80 d}-\frac {7}{16} b^2 x \left (240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right )+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2734
Rule 2753
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\int (a+b \sin (c+d x))^6 \left (7 b^2+7 a b \sin (c+d x)\right ) \, dx\\ &=\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{7} \int (a+b \sin (c+d x))^5 \left (91 a b^2+7 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{42} \int (a+b \sin (c+d x))^4 \left (7 b^2 \left (108 a^2+35 b^2\right )+7 a b \left (30 a^2+113 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{210} \int (a+b \sin (c+d x))^3 \left (231 a b^2 \left (20 a^2+19 b^2\right )+7 b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {1}{840} \int (a+b \sin (c+d x))^2 \left (21 b^2 \left (1000 a^4+1828 a^2 b^2+175 b^4\right )+63 a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}-\frac {\int (a+b \sin (c+d x)) \left (63 a b^2 \left (1080 a^4+3076 a^2 b^2+849 b^4\right )+63 b \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \sin (c+d x)\right ) \, dx}{2520}\\ &=-\frac {7}{16} b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) x+\frac {a b \left (40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right ) \cos (c+d x)}{20 d}+\frac {b^2 \left (80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right ) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a b \left (40 a^4+624 a^2 b^2+337 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {b \left (120 a^4+992 a^2 b^2+175 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac {a b \left (30 a^2+113 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac {b \left (6 a^2+7 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac {a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac {\sec (c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{d}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 313, normalized size = 0.90 \[ \frac {\sec (c+d x) \left (1920 a^8 \sin (c+d x)+15360 a^7 b+53760 a^6 b^2 \sin (c+d x)+161280 a^5 b^3+151200 a^4 b^4 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))+201600 a^3 b^5+67200 a^2 b^6 \sin (c+d x)+12600 a^2 b^6 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))+1120 \left (48 a^5 b^3+80 a^3 b^5+15 a b^7\right ) \cos (2 (c+d x))-840 b^2 \left (64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right ) (c+d x) \cos (c+d x)-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+33600 a b^7+2625 b^8 \sin (c+d x)+630 b^8 \sin (3 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right )}{1920 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 266, normalized size = 0.76 \[ \frac {384 \, a b^{7} \cos \left (d x + c\right )^{6} + 1920 \, a^{7} b + 13440 \, a^{5} b^{3} + 13440 \, a^{3} b^{5} + 1920 \, a b^{7} - 640 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 105 \, {\left (64 \, a^{6} b^{2} + 240 \, a^{4} b^{4} + 120 \, a^{2} b^{6} + 5 \, b^{8}\right )} d x \cos \left (d x + c\right ) + 1920 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, b^{8} \cos \left (d x + c\right )^{6} + 48 \, a^{8} + 1344 \, a^{6} b^{2} + 3360 \, a^{4} b^{4} + 1344 \, a^{2} b^{6} + 48 \, b^{8} - 2 \, {\left (168 \, a^{2} b^{6} + 19 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (560 \, a^{4} b^{4} + 504 \, a^{2} b^{6} + 29 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 799, normalized size = 2.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 406, normalized size = 1.16 \[ \frac {a^{8} \tan \left (d x +c \right )+\frac {8 a^{7} b}{\cos \left (d x +c \right )}+28 a^{6} b^{2} \left (\tan \left (d x +c \right )-d x -c \right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+28 a^{2} b^{6} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+b^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 348, normalized size = 1.00 \[ -\frac {6720 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{6} b^{2} + 8400 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{4} b^{4} + 4480 \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{3} b^{5} + 840 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2} b^{6} - 384 \, {\left (\cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3} + \frac {5}{\cos \left (d x + c\right )} + 15 \, \cos \left (d x + c\right )\right )} a b^{7} + 5 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} b^{8} - 13440 \, a^{5} b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 240 \, a^{8} \tan \left (d x + c\right ) - \frac {1920 \, a^{7} b}{\cos \left (d x + c\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.73, size = 767, normalized size = 2.20 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (240\,a^7\,b+1120\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (96\,a^7\,b+224\,a^5\,b^3\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+\frac {256\,a\,b^7}{5}+16\,a^7\,b+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (96\,a^7\,b+1120\,a^5\,b^3+\frac {4480\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (240\,a^7\,b+2240\,a^5\,b^3+2688\,a^3\,b^5+\frac {2304\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (320\,a^7\,b+2240\,a^5\,b^3+\frac {6272\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8+56\,a^6\,b^2+210\,a^4\,b^4+105\,a^2\,b^6+\frac {35\,b^8}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (12\,a^8+336\,a^6\,b^2+980\,a^4\,b^4+490\,a^2\,b^6+\frac {245\,b^8}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (30\,a^8+840\,a^6\,b^2+2030\,a^4\,b^4+791\,a^2\,b^6+\frac {791\,b^8}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (40\,a^8+1120\,a^6\,b^2+2520\,a^4\,b^4+812\,a^2\,b^6+\frac {25\,b^8}{2}\right )+\frac {896\,a^3\,b^5}{3}+224\,a^5\,b^3+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,b^2\,\mathrm {atan}\left (\frac {7\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{448\,a^6\,b^2+1680\,a^4\,b^4+840\,a^2\,b^6+35\,b^8}\right )\,\left (64\,a^6+240\,a^4\,b^2+120\,a^2\,b^4+5\,b^6\right )}{8\,d} \]
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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