Integrand size = 21, antiderivative size = 267 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d} \]
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Time = 0.60 (sec) , antiderivative size = 267, 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} \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (a \cos (c+d x)+b)}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d} \]
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Rule 12
Rule 962
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^7(c+d x)}{(-b-a \cos (c+d x))^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^3}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^3}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2-\frac {b^2 \left (-a^2+b^2\right )^3}{(b-x)^2}+\frac {2 b \left (-a^2+b^2\right )^2 \left (-a^2+4 b^2\right )}{b-x}-6 b \left (-a^2+b^2\right )^2 x-\left (3 a^4-9 a^2 b^2+5 b^4\right ) x^2-2 b \left (-3 a^2+2 b^2\right ) x^3+3 \left (a^2-b^2\right ) x^4-2 b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^9 d} \\ & = -\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d} \\ \end{align*}
Time = 3.19 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-3675 a^8+61320 a^6 b^2-132720 a^4 b^4+87360 a^2 b^6-13440 b^8-140 \left (21 a^8-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6\right ) \cos (2 (c+d x))-3780 a^7 b \cos (3 (c+d x))+8400 a^5 b^3 \cos (3 (c+d x))-4480 a^3 b^5 \cos (3 (c+d x))+588 a^8 \cos (4 (c+d x))-1848 a^6 b^2 \cos (4 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))+476 a^7 b \cos (5 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))-132 a^8 \cos (6 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))-40 a^7 b \cos (7 (c+d x))+15 a^8 \cos (8 (c+d x))+26880 a^6 b^2 \log (b+a \cos (c+d x))-161280 a^4 b^4 \log (b+a \cos (c+d x))+241920 a^2 b^6 \log (b+a \cos (c+d x))-107520 b^8 \log (b+a \cos (c+d x))+1680 a b \cos (c+d x) \left (-8 a^6+67 a^4 b^2-116 a^2 b^4+56 b^6+16 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))\right )}{13440 a^9 d (b+a \cos (c+d x))} \]
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Time = 2.09 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{2}-\cos \left (d x +c \right )^{4} a^{3} b^{3}+\cos \left (d x +c \right )^{3} a^{6}-3 \cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {5 \cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-3 \cos \left (d x +c \right )^{2} a^{5} b +6 \cos \left (d x +c \right )^{2} a^{3} b^{3}-3 \cos \left (d x +c \right )^{2} a \,b^{5}-\cos \left (d x +c \right ) a^{6}+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 b^{6} \cos \left (d x +c \right )}{a^{8}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}}{d}\) | \(321\) |
default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{2}-\cos \left (d x +c \right )^{4} a^{3} b^{3}+\cos \left (d x +c \right )^{3} a^{6}-3 \cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {5 \cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-3 \cos \left (d x +c \right )^{2} a^{5} b +6 \cos \left (d x +c \right )^{2} a^{3} b^{3}-3 \cos \left (d x +c \right )^{2} a \,b^{5}-\cos \left (d x +c \right ) a^{6}+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 b^{6} \cos \left (d x +c \right )}{a^{8}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}}{d}\) | \(321\) |
parallelrisch | \(\frac {2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-11 \left (\frac {245 a \left (a^{6}-\frac {76}{7} a^{4} b^{2}+\frac {400}{21} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (2 d x +2 c \right )}{11}+\frac {35 \left (9 b \,a^{6}-20 a^{4} b^{3}+\frac {32}{3} b^{5} a^{2}\right ) \cos \left (3 d x +3 c \right )}{11}+7 \left (2 a^{5} b^{2}-\frac {40}{33} b^{4} a^{3}-\frac {7}{11} a^{7}\right ) \cos \left (4 d x +4 c \right )+\frac {7 \left (-\frac {17}{3} b \,a^{6}+4 a^{4} b^{3}\right ) \cos \left (5 d x +5 c \right )}{11}+\left (-\frac {28}{33} a^{5} b^{2}+a^{7}\right ) \cos \left (6 d x +6 c \right )+\frac {10 a^{6} b \cos \left (7 d x +7 c \right )}{33}-\frac {5 a^{7} \cos \left (8 d x +8 c \right )}{44}+\frac {4 \left (-2128 a^{5} b^{2}+\frac {12880}{3} b^{4} a^{3}+\frac {175}{3} b \,a^{6}-1960 a^{4} b^{3}+4200 b^{5} a^{2}-2240 b^{7}+128 a^{7}-2240 a \,b^{6}\right ) \cos \left (d x +c \right )}{11}+\frac {1225 a^{7}}{44}-\frac {6720 a \,b^{6}}{11}+\frac {12600 b^{4} a^{3}}{11}+\frac {51520 b^{5} a^{2}}{33}-\frac {8960 b^{7}}{11}+\frac {512 b \,a^{6}}{11}-\frac {17990 a^{5} b^{2}}{33}-\frac {8512 a^{4} b^{3}}{11}\right ) a}{1120 \left (b +a \cos \left (d x +c \right )\right ) a^{9} d}\) | \(428\) |
risch | \(\frac {7 \cos \left (3 d x +3 c \right )}{64 d \,a^{2}}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 a^{2} d}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 a^{2} d}-\frac {2 b^{2} \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{9} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {24 i b^{3} c}{a^{5} d}-\frac {36 i b^{5} c}{a^{7} d}+\frac {16 i b^{7} c}{a^{9} d}-\frac {7 \cos \left (5 d x +5 c \right )}{320 d \,a^{2}}-\frac {b \cos \left (6 d x +6 c \right )}{96 d \,a^{3}}+\frac {3 \cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{4}}+\frac {b \cos \left (4 d x +4 c \right )}{8 a^{3} d}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{8 a^{5} d}-\frac {9 \cos \left (3 d x +3 c \right ) b^{2}}{16 a^{4} d}+\frac {5 \cos \left (3 d x +3 c \right ) b^{4}}{12 a^{6} d}-\frac {4 i b c}{a^{3} d}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {57 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{8} d}+\frac {57 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{8} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}-\frac {12 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{5} d}+\frac {18 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{7} d}-\frac {8 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{9} d}-\frac {2 i b x}{a^{3}}+\frac {12 i b^{3} x}{a^{5}}-\frac {18 i b^{5} x}{a^{7}}+\frac {8 i b^{7} x}{a^{9}}\) | \(749\) |
norman | \(\frac {\frac {\left (96 a^{7}+96 b \,a^{6}+380 a^{5} b^{2}-1596 a^{4} b^{3}-1188 b^{4} a^{3}+3220 b^{5} a^{2}+720 a \,b^{6}-1680 b^{7}\right ) \left (a +b \right )}{210 a^{8} d b}-\frac {\left (32 a^{8}-160 a^{7} b -32 a^{6} b^{2}+172 a^{5} b^{3}-644 a^{4} b^{4}-196 a^{3} b^{5}+1180 a^{2} b^{6}+80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5 a^{8} d b}+\frac {\left (32 a^{8}+160 a^{7} b -32 a^{6} b^{2}-172 a^{5} b^{3}-644 a^{4} b^{4}+196 a^{3} b^{5}+1180 a^{2} b^{6}-80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5 a^{8} d b}-\frac {\left (96 a^{8}-288 a^{7} b +212 a^{6} b^{2}+896 a^{5} b^{3}-2160 a^{4} b^{4}-1144 a^{3} b^{5}+3620 a^{2} b^{6}+480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{15 a^{8} d b}-\frac {\left (96 a^{8}-192 a^{7} b +476 a^{6} b^{2}+1216 a^{5} b^{3}-2784 a^{4} b^{4}-2032 a^{3} b^{5}+3940 a^{2} b^{6}+960 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{210 d \,a^{8} b}+\frac {\left (96 a^{8}+288 a^{7} b +212 a^{6} b^{2}-896 a^{5} b^{3}-2160 a^{4} b^{4}+1144 a^{3} b^{5}+3620 a^{2} b^{6}-480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 a^{8} d b}-\frac {\left (288 a^{8}-672 a^{7} b +1104 a^{6} b^{2}+3268 a^{5} b^{3}-7428 a^{4} b^{4}-4908 a^{3} b^{5}+11260 a^{2} b^{6}+2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{105 a^{8} d b}+\frac {\left (288 a^{8}+672 a^{7} b +1104 a^{6} b^{2}-3268 a^{5} b^{3}-7428 a^{4} b^{4}+4908 a^{3} b^{5}+11260 a^{2} b^{6}-2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{105 d \,a^{8} b}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{9} d}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{9} d}\) | \(836\) |
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Time = 0.35 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {120 \, a^{8} \cos \left (d x + c\right )^{8} - 160 \, a^{7} b \cos \left (d x + c\right )^{7} + 1715 \, a^{6} b^{2} - 4725 \, a^{4} b^{4} + 3780 \, a^{2} b^{6} - 840 \, b^{8} - 56 \, {\left (9 \, a^{8} - 4 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{6} + 84 \, {\left (9 \, a^{7} b - 4 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (6 \, a^{8} - 9 \, a^{6} b^{2} + 4 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 280 \, {\left (6 \, a^{7} b - 9 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 840 \, {\left (a^{8} - 6 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (a^{7} b + 153 \, a^{5} b^{3} - 324 \, a^{3} b^{5} + 168 \, a b^{7}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{2} - 6 \, a^{4} b^{4} + 9 \, a^{2} b^{6} - 4 \, b^{8} + {\left (a^{7} b - 6 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{840 \, {\left (a^{10} d \cos \left (d x + c\right ) + a^{9} b d\right )}} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {210 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )}}{a^{10} \cos \left (d x + c\right ) + a^{9} b} + \frac {30 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 126 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - 2 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (3 \, a^{6} - 9 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 210 \, {\left (a^{6} - 9 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{8}} + \frac {420 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{9}}}{210 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (257) = 514\).
Time = 0.37 (sec) , antiderivative size = 1861, normalized size of antiderivative = 6.97 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 0.19 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.20 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {b^3}{2\,a^5}+\frac {b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{2\,a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{2\,a^2}+\frac {b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^2}-\frac {3\,b^2}{5\,a^4}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a^2}-\frac {2\,b\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a^2}+\frac {2\,b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{3\,a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{3\,a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{3\,a^3\,d}-\frac {-a^6\,b^2+3\,a^4\,b^4-3\,a^2\,b^6+b^8}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^9+b\,a^8\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^6\,b-12\,a^4\,b^3+18\,a^2\,b^5-8\,b^7\right )}{a^9\,d} \]
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