Integrand size = 21, antiderivative size = 329 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (b+a \cos (c+d x))^2}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (b+a \cos (c+d x))}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (b+a \cos (c+d x))}{a^{10} d} \]
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Time = 0.58 (sec) , antiderivative size = 329, 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 = {3957, 2916, 12, 962} \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (a \cos (c+d x)+b)}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (a \cos (c+d x)+b)^2}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (a \cos (c+d x)+b)}{a^{10} d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}-\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d} \]
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Rule 12
Rule 962
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )^3}{a^3 (-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )^3}{(-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^{10} d} \\ & = \frac {\text {Subst}\left (\int \left (a^6 \left (1+\frac {-18 a^4 b^2+45 a^2 b^4-28 b^6}{a^6}\right )+\frac {b^3 \left (-a^2+b^2\right )^3}{(b-x)^3}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{(b-x)^2}+\frac {3 b \left (-a^6+10 a^4 b^2-21 a^2 b^4+12 b^6\right )}{b-x}-3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) x-3 \left (a^4-6 a^2 b^2+5 b^4\right ) x^2-b \left (-9 a^2+10 b^2\right ) x^3+3 \left (a^2-2 b^2\right ) x^4-3 b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{10} d} \\ & = -\frac {\left (a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right ) \cos (c+d x)}{a^9 d}-\frac {3 b \left (3 a^4-10 a^2 b^2+7 b^4\right ) \cos ^2(c+d x)}{2 a^8 d}+\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^7 d}+\frac {b \left (9 a^2-10 b^2\right ) \cos ^4(c+d x)}{4 a^6 d}-\frac {3 \left (a^2-2 b^2\right ) \cos ^5(c+d x)}{5 a^5 d}-\frac {b \cos ^6(c+d x)}{2 a^4 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {b^3 \left (a^2-b^2\right )^3}{2 a^{10} d (b+a \cos (c+d x))^2}+\frac {3 b^2 \left (a^2-3 b^2\right ) \left (a^2-b^2\right )^2}{a^{10} d (b+a \cos (c+d x))}+\frac {3 b \left (a^2-b^2\right ) \left (a^4-9 a^2 b^2+12 b^4\right ) \log (b+a \cos (c+d x))}{a^{10} d} \\ \end{align*}
Time = 5.50 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.67 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-7945 a^8 b+164080 a^6 b^3-502320 a^4 b^5+425600 a^2 b^7-76160 b^9-784 a^9 \cos (3 (c+d x))+17528 a^7 b^2 \cos (3 (c+d x))-43680 a^5 b^4 \cos (3 (c+d x))+26880 a^3 b^6 \cos (3 (c+d x))-1456 a^8 b \cos (4 (c+d x))+4872 a^6 b^3 \cos (4 (c+d x))-3360 a^4 b^5 \cos (4 (c+d x))+152 a^9 \cos (5 (c+d x))-840 a^7 b^2 \cos (5 (c+d x))+672 a^5 b^4 \cos (5 (c+d x))+174 a^8 b \cos (6 (c+d x))-168 a^6 b^3 \cos (6 (c+d x))-39 a^9 \cos (7 (c+d x))+48 a^7 b^2 \cos (7 (c+d x))-15 a^8 b \cos (8 (c+d x))+5 a^9 \cos (9 (c+d x))+13440 a^8 b \log (b+a \cos (c+d x))-107520 a^6 b^3 \log (b+a \cos (c+d x))+13440 a^4 b^5 \log (b+a \cos (c+d x))+403200 a^2 b^7 \log (b+a \cos (c+d x))-322560 b^9 \log (b+a \cos (c+d x))+70 a^2 b \cos (2 (c+d x)) \left (-137 a^6+1896 a^4 b^2-4656 a^2 b^4+2912 b^6+192 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )-70 a \cos (c+d x) \left (49 a^8-1472 a^6 b^2+3216 a^4 b^4+576 a^2 b^6-2432 b^8-768 b^2 \left (a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right ) \log (b+a \cos (c+d x))\right )}{8960 a^{10} d (b+a \cos (c+d x))^2} \]
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Time = 2.89 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{2}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {6 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {9 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {5 \cos \left (d x +c \right )^{4} a^{3} b^{3}}{2}+\cos \left (d x +c \right )^{3} a^{6}-6 \cos \left (d x +c \right )^{3} a^{4} b^{2}+5 \cos \left (d x +c \right )^{3} a^{2} b^{4}-\frac {9 \cos \left (d x +c \right )^{2} a^{5} b}{2}+15 \cos \left (d x +c \right )^{2} a^{3} b^{3}-\frac {21 \cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+18 \cos \left (d x +c \right ) a^{4} b^{2}-45 \cos \left (d x +c \right ) a^{2} b^{4}+28 b^{6} \cos \left (d x +c \right )}{a^{9}}+\frac {3 b^{2} \left (a^{6}-5 a^{4} b^{2}+7 a^{2} b^{4}-3 b^{6}\right )}{a^{10} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{2 a^{10} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 b \left (a^{6}-10 a^{4} b^{2}+21 a^{2} b^{4}-12 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{10}}}{d}\) | \(367\) |
default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{2}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {6 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {9 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {5 \cos \left (d x +c \right )^{4} a^{3} b^{3}}{2}+\cos \left (d x +c \right )^{3} a^{6}-6 \cos \left (d x +c \right )^{3} a^{4} b^{2}+5 \cos \left (d x +c \right )^{3} a^{2} b^{4}-\frac {9 \cos \left (d x +c \right )^{2} a^{5} b}{2}+15 \cos \left (d x +c \right )^{2} a^{3} b^{3}-\frac {21 \cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+18 \cos \left (d x +c \right ) a^{4} b^{2}-45 \cos \left (d x +c \right ) a^{2} b^{4}+28 b^{6} \cos \left (d x +c \right )}{a^{9}}+\frac {3 b^{2} \left (a^{6}-5 a^{4} b^{2}+7 a^{2} b^{4}-3 b^{6}\right )}{a^{10} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{3} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{2 a^{10} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 b \left (a^{6}-10 a^{4} b^{2}+21 a^{2} b^{4}-12 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{10}}}{d}\) | \(367\) |
parallelrisch | \(\frac {1680 b \left (a -b \right ) \left (a +b \right ) \left (a^{4}-9 a^{2} b^{2}+12 b^{4}\right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-1680 b \left (a -b \right ) \left (a +b \right ) \left (a^{4}-9 a^{2} b^{2}+12 b^{4}\right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+19 a \left (-\frac {256 \left (a^{7}+\frac {2135}{1024} b \,a^{6}-\frac {567}{16} a^{5} b^{2}-\frac {15225}{256} a^{4} b^{3}+\frac {1785}{16} b^{4} a^{3}+\frac {5565}{32} b^{5} a^{2}-\frac {315}{4} a \,b^{6}-\frac {945}{8} b^{7}\right ) a \cos \left (2 d x +2 c \right )}{19}-\frac {98 a^{2} \left (a^{6}-\frac {313}{14} a^{4} b^{2}+\frac {390}{7} a^{2} b^{4}-\frac {240}{7} b^{6}\right ) \cos \left (3 d x +3 c \right )}{19}-\frac {182 a^{3} b \left (a^{4}-\frac {87}{26} a^{2} b^{2}+\frac {30}{13} b^{4}\right ) \cos \left (4 d x +4 c \right )}{19}+a^{4} \left (a^{4}-\frac {105}{19} a^{2} b^{2}+\frac {84}{19} b^{4}\right ) \cos \left (5 d x +5 c \right )+\frac {3 \left (\frac {29}{4} a^{7} b -7 a^{5} b^{3}\right ) \cos \left (6 d x +6 c \right )}{19}+\frac {3 \left (-\frac {13}{8} a^{8}+2 a^{6} b^{2}\right ) \cos \left (7 d x +7 c \right )}{19}-\frac {15 \cos \left (8 d x +8 c \right ) a^{7} b}{152}+\frac {5 a^{8} \cos \left (9 d x +9 c \right )}{152}+\frac {\left (40320 b^{8}-\frac {1715}{4} a^{8}-1024 a^{7} b +15540 a^{6} b^{2}+36288 a^{5} b^{3}-33600 a^{4} b^{4}-114240 a^{3} b^{5}-20160 a^{2} b^{6}+80640 a \,b^{7}\right ) \cos \left (d x +c \right )}{19}+\frac {40320 b^{8}}{19}-\frac {36960 a^{2} b^{6}}{19}-\frac {2625 a^{7} b}{152}+\frac {50400 a \,b^{7}}{19}-\frac {256 a^{8}}{19}+\frac {8560 a^{6} b^{2}}{19}+\frac {20475 a^{5} b^{3}}{19}-\frac {10416 a^{4} b^{4}}{19}-\frac {69300 a^{3} b^{5}}{19}\right )}{560 \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) a^{10} d}\) | \(581\) |
risch | \(\text {Expression too large to display}\) | \(962\) |
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Time = 0.37 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.36 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {80 \, a^{9} \cos \left (d x + c\right )^{9} - 120 \, a^{8} b \cos \left (d x + c\right )^{8} + 2275 \, a^{6} b^{3} - 11235 \, a^{4} b^{5} + 13860 \, a^{2} b^{7} - 4760 \, b^{9} - 48 \, {\left (7 \, a^{9} - 4 \, a^{7} b^{2}\right )} \cos \left (d x + c\right )^{7} + 84 \, {\left (7 \, a^{8} b - 4 \, a^{6} b^{3}\right )} \cos \left (d x + c\right )^{6} + 56 \, {\left (10 \, a^{9} - 21 \, a^{7} b^{2} + 12 \, a^{5} b^{4}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (10 \, a^{8} b - 21 \, a^{6} b^{3} + 12 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )^{4} - 560 \, {\left (a^{9} - 10 \, a^{7} b^{2} + 21 \, a^{5} b^{4} - 12 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 35 \, {\left (7 \, a^{8} b - 399 \, a^{6} b^{3} + 1116 \, a^{4} b^{5} - 728 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 70 \, {\left (41 \, a^{7} b^{2} - 81 \, a^{5} b^{4} - 108 \, a^{3} b^{6} + 152 \, a b^{8}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{3} - 10 \, a^{4} b^{5} + 21 \, a^{2} b^{7} - 12 \, b^{9} + {\left (a^{8} b - 10 \, a^{6} b^{3} + 21 \, a^{4} b^{5} - 12 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{2} - 10 \, a^{5} b^{4} + 21 \, a^{3} b^{6} - 12 \, a b^{8}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{560 \, {\left (a^{12} d \cos \left (d x + c\right )^{2} + 2 \, a^{11} b d \cos \left (d x + c\right ) + a^{10} b^{2} d\right )}} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {70 \, {\left (5 \, a^{6} b^{3} - 27 \, a^{4} b^{5} + 39 \, a^{2} b^{7} - 17 \, b^{9} + 6 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 7 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )\right )}}{a^{12} \cos \left (d x + c\right )^{2} + 2 \, a^{11} b \cos \left (d x + c\right ) + a^{10} b^{2}} + \frac {20 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (a^{6} - 2 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 35 \, {\left (9 \, a^{5} b - 10 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{6} - 6 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} - 140 \, {\left (a^{6} - 18 \, a^{4} b^{2} + 45 \, a^{2} b^{4} - 28 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{9}} + \frac {420 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - 12 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10}}}{140 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2150 vs. \(2 (317) = 634\).
Time = 0.46 (sec) , antiderivative size = 2150, normalized size of antiderivative = 6.53 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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Time = 0.23 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.32 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {2\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{4\,a}\right )}{d}-\frac {\frac {-5\,a^6\,b^3+27\,a^4\,b^5-39\,a^2\,b^7+17\,b^9}{2\,a}+\cos \left (c+d\,x\right )\,\left (-3\,a^6\,b^2+15\,a^4\,b^4-21\,a^2\,b^6+9\,b^8\right )}{d\,\left (a^{11}\,{\cos \left (c+d\,x\right )}^2+2\,a^{10}\,b\,\cos \left (c+d\,x\right )+a^9\,b^2\right )}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^3}-\frac {6\,b^2}{5\,a^5}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,b^2\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{2\,a^2}-\frac {b^3\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{2\,a^3}+\frac {3\,b\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{2\,a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^3\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^3}+\frac {b^3\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a^3}+\frac {3\,b^2\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{a^2}-\frac {3\,b\,\left (\frac {3\,b^2\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a^2}-\frac {b^3\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^3}+\frac {3\,b\,\left (\frac {3}{a^3}+\frac {3\,b^4}{a^7}+\frac {3\,b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {3\,b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^3}+\frac {b^4}{a^7}+\frac {b^2\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a^2}-\frac {b\,\left (\frac {8\,b^3}{a^6}+\frac {3\,b\,\left (\frac {3}{a^3}-\frac {6\,b^2}{a^5}\right )}{a}\right )}{a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{2\,a^4\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^6\,b-30\,a^4\,b^3+63\,a^2\,b^5-36\,b^7\right )}{a^{10}\,d} \]
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