Integrand size = 31, antiderivative size = 236 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\frac {7 (4 A+B) \text {arctanh}(\sin (c+d x))}{128 a^2 d}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{80 d (a+a \sin (c+d x))^5}-\frac {a^2 (2 A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a (5 A-B)}{96 d (a+a \sin (c+d x))^3}-\frac {5 A}{64 d (a+a \sin (c+d x))^2}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2+a^2 \sin (c+d x)\right )} \]
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Time = 0.20 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2915, 78, 212} \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=-\frac {a^3 (A-B)}{80 d (a \sin (c+d x)+a)^5}+\frac {7 (4 A+B) \text {arctanh}(\sin (c+d x))}{128 a^2 d}-\frac {a^2 (2 A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}-\frac {a (5 A-B)}{96 d (a \sin (c+d x)+a)^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {5 A}{64 d (a \sin (c+d x)+a)^2} \]
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Rule 78
Rule 212
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {a^7 \text {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^4 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^7 \text {Subst}\left (\int \left (\frac {A+B}{64 a^6 (a-x)^4}+\frac {3 A+2 B}{64 a^7 (a-x)^3}+\frac {3 (7 A+3 B)}{256 a^8 (a-x)^2}+\frac {A-B}{16 a^4 (a+x)^6}+\frac {2 A-B}{16 a^5 (a+x)^5}+\frac {5 A-B}{32 a^6 (a+x)^4}+\frac {5 A}{32 a^7 (a+x)^3}+\frac {5 (7 A+B)}{256 a^8 (a+x)^2}+\frac {7 (4 A+B)}{128 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{80 d (a+a \sin (c+d x))^5}-\frac {a^2 (2 A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a (5 A-B)}{96 d (a+a \sin (c+d x))^3}-\frac {5 A}{64 d (a+a \sin (c+d x))^2}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {(7 (4 A+B)) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a d} \\ & = \frac {7 (4 A+B) \text {arctanh}(\sin (c+d x))}{128 a^2 d}+\frac {a (A+B)}{192 d (a-a \sin (c+d x))^3}+\frac {3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{80 d (a+a \sin (c+d x))^5}-\frac {a^2 (2 A-B)}{64 d (a+a \sin (c+d x))^4}-\frac {a (5 A-B)}{96 d (a+a \sin (c+d x))^3}-\frac {5 A}{64 d (a+a \sin (c+d x))^2}+\frac {3 (7 A+3 B)}{256 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {5 (7 A+B)}{256 d \left (a^2+a^2 \sin (c+d x)\right )} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.68 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\frac {210 (4 A+B) \text {arctanh}(\sin (c+d x))-\frac {2 \left (48 (-8 A+3 B)+183 (4 A+B) \sin (c+d x)+462 (4 A+B) \sin ^2(c+d x)-49 (4 A+B) \sin ^3(c+d x)-560 (4 A+B) \sin ^4(c+d x)-175 (4 A+B) \sin ^5(c+d x)+210 (4 A+B) \sin ^6(c+d x)+105 (4 A+B) \sin ^7(c+d x)\right )}{(-1+\sin (c+d x))^3 (1+\sin (c+d x))^5}}{3840 a^2 d} \]
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Time = 2.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {-\frac {5 A}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {\frac {A}{16}-\frac {B}{16}}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {\frac {A}{8}-\frac {B}{16}}{4 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {\frac {5 A}{32}-\frac {B}{32}}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\left (\frac {7 A}{64}+\frac {7 B}{256}\right ) \ln \left (1+\sin \left (d x +c \right )\right )-\frac {\frac {35 A}{256}+\frac {5 B}{256}}{1+\sin \left (d x +c \right )}+\left (-\frac {7 A}{64}-\frac {7 B}{256}\right ) \ln \left (\sin \left (d x +c \right )-1\right )-\frac {-\frac {3 A}{64}-\frac {B}{32}}{2 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {\frac {A}{64}+\frac {B}{64}}{3 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {\frac {21 A}{256}+\frac {9 B}{256}}{\sin \left (d x +c \right )-1}}{d \,a^{2}}\) | \(189\) |
default | \(\frac {-\frac {5 A}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {\frac {A}{16}-\frac {B}{16}}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {\frac {A}{8}-\frac {B}{16}}{4 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {\frac {5 A}{32}-\frac {B}{32}}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\left (\frac {7 A}{64}+\frac {7 B}{256}\right ) \ln \left (1+\sin \left (d x +c \right )\right )-\frac {\frac {35 A}{256}+\frac {5 B}{256}}{1+\sin \left (d x +c \right )}+\left (-\frac {7 A}{64}-\frac {7 B}{256}\right ) \ln \left (\sin \left (d x +c \right )-1\right )-\frac {-\frac {3 A}{64}-\frac {B}{32}}{2 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {\frac {A}{64}+\frac {B}{64}}{3 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {\frac {21 A}{256}+\frac {9 B}{256}}{\sin \left (d x +c \right )-1}}{d \,a^{2}}\) | \(189\) |
parallelrisch | \(\frac {-420 \left (A +\frac {B}{4}\right ) \left (-20 \cos \left (4 d x +4 c \right )+\cos \left (8 d x +8 c \right )-4 \sin \left (7 d x +7 c \right )-20 \sin \left (5 d x +5 c \right )-64 \cos \left (2 d x +2 c \right )-36 \sin \left (3 d x +3 c \right )-45-20 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+420 \left (A +\frac {B}{4}\right ) \left (-20 \cos \left (4 d x +4 c \right )+\cos \left (8 d x +8 c \right )-4 \sin \left (7 d x +7 c \right )-20 \sin \left (5 d x +5 c \right )-64 \cos \left (2 d x +2 c \right )-36 \sin \left (3 d x +3 c \right )-45-20 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (736 A +15544 B \right ) \cos \left (2 d x +2 c \right )+\left (8000 A +6800 B \right ) \cos \left (4 d x +4 c \right )+\left (384 A -144 B \right ) \cos \left (8 d x +8 c \right )+\left (-30456 A +1026 B \right ) \sin \left (3 d x +3 c \right )+\left (-7960 A +2810 B \right ) \sin \left (5 d x +5 c \right )+\left (-696 A +786 B \right ) \sin \left (7 d x +7 c \right )+\left (3360 A +840 B \right ) \cos \left (6 d x +6 c \right )+\left (-55960 A -9190 B \right ) \sin \left (d x +c \right )-12480 A -23040 B}{1920 d \,a^{2} \left (-20 \cos \left (4 d x +4 c \right )+\cos \left (8 d x +8 c \right )-4 \sin \left (7 d x +7 c \right )-20 \sin \left (5 d x +5 c \right )-64 \cos \left (2 d x +2 c \right )-36 \sin \left (3 d x +3 c \right )-45-20 \sin \left (d x +c \right )\right )}\) | \(411\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (420 A \,{\mathrm e}^{14 i \left (d x +c \right )}-420 A -105 B +1960 i B \,{\mathrm e}^{11 i \left (d x +c \right )}+7840 i A \,{\mathrm e}^{11 i \left (d x +c \right )}-8316 A \,{\mathrm e}^{10 i \left (d x +c \right )}-2079 B \,{\mathrm e}^{10 i \left (d x +c \right )}+3164 i B \,{\mathrm e}^{9 i \left (d x +c \right )}+12656 i A \,{\mathrm e}^{9 i \left (d x +c \right )}+8316 A \,{\mathrm e}^{4 i \left (d x +c \right )}+2079 B \,{\mathrm e}^{4 i \left (d x +c \right )}+140 A \,{\mathrm e}^{2 i \left (d x +c \right )}+35 B \,{\mathrm e}^{2 i \left (d x +c \right )}+420 i B \,{\mathrm e}^{13 i \left (d x +c \right )}+1680 i A \,{\mathrm e}^{13 i \left (d x +c \right )}+7840 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+1960 i B \,{\mathrm e}^{3 i \left (d x +c \right )}+1680 i A \,{\mathrm e}^{i \left (d x +c \right )}+420 i B \,{\mathrm e}^{i \left (d x +c \right )}-24140 A \,{\mathrm e}^{8 i \left (d x +c \right )}-6035 B \,{\mathrm e}^{8 i \left (d x +c \right )}+24140 A \,{\mathrm e}^{6 i \left (d x +c \right )}+6035 B \,{\mathrm e}^{6 i \left (d x +c \right )}-29520 i B \,{\mathrm e}^{7 i \left (d x +c \right )}+12656 i A \,{\mathrm e}^{5 i \left (d x +c \right )}+3164 i B \,{\mathrm e}^{5 i \left (d x +c \right )}+4800 i A \,{\mathrm e}^{7 i \left (d x +c \right )}-35 B \,{\mathrm e}^{12 i \left (d x +c \right )}+105 B \,{\mathrm e}^{14 i \left (d x +c \right )}-140 A \,{\mathrm e}^{12 i \left (d x +c \right )}\right )}{960 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d \,a^{2}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{32 a^{2} d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{128 a^{2} d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{32 a^{2} d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{128 a^{2} d}\) | \(545\) |
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Time = 0.32 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\frac {420 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} - 140 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{4} - 56 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left ({\left (4 \, A + B\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (4 \, A + B\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (105 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{6} - 140 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{4} - 84 \, {\left (4 \, A + B\right )} \cos \left (d x + c\right )^{2} - 256 \, A - 64 \, B\right )} \sin \left (d x + c\right ) - 128 \, A - 512 \, B}{3840 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} - 2 \, a^{2} d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{6}\right )}} \]
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Timed out. \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (105 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{7} + 210 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{6} - 175 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{5} - 560 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{4} - 49 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{3} + 462 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right )^{2} + 183 \, {\left (4 \, A + B\right )} \sin \left (d x + c\right ) - 384 \, A + 144 \, B\right )}}{a^{2} \sin \left (d x + c\right )^{8} + 2 \, a^{2} \sin \left (d x + c\right )^{7} - 2 \, a^{2} \sin \left (d x + c\right )^{6} - 6 \, a^{2} \sin \left (d x + c\right )^{5} + 6 \, a^{2} \sin \left (d x + c\right )^{3} + 2 \, a^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - a^{2}} - \frac {105 \, {\left (4 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {105 \, {\left (4 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{3840 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {420 \, {\left (4 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {420 \, {\left (4 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} + \frac {10 \, {\left (308 \, A \sin \left (d x + c\right )^{3} + 77 \, B \sin \left (d x + c\right )^{3} - 1050 \, A \sin \left (d x + c\right )^{2} - 285 \, B \sin \left (d x + c\right )^{2} + 1212 \, A \sin \left (d x + c\right ) + 363 \, B \sin \left (d x + c\right ) - 478 \, A - 163 \, B\right )}}{a^{2} {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {3836 \, A \sin \left (d x + c\right )^{5} + 959 \, B \sin \left (d x + c\right )^{5} + 21280 \, A \sin \left (d x + c\right )^{4} + 5095 \, B \sin \left (d x + c\right )^{4} + 47960 \, A \sin \left (d x + c\right )^{3} + 10790 \, B \sin \left (d x + c\right )^{3} + 55360 \, A \sin \left (d x + c\right )^{2} + 11230 \, B \sin \left (d x + c\right )^{2} + 33260 \, A \sin \left (d x + c\right ) + 5435 \, B \sin \left (d x + c\right ) + 8608 \, A + 667 \, B}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{15360 \, d} \]
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Time = 10.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.02 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\frac {7\,A}{32}+\frac {7\,B}{128}\right )\,{\sin \left (c+d\,x\right )}^7+\left (\frac {7\,A}{16}+\frac {7\,B}{64}\right )\,{\sin \left (c+d\,x\right )}^6+\left (-\frac {35\,A}{96}-\frac {35\,B}{384}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {7\,A}{6}-\frac {7\,B}{24}\right )\,{\sin \left (c+d\,x\right )}^4+\left (-\frac {49\,A}{480}-\frac {49\,B}{1920}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {77\,A}{80}+\frac {77\,B}{320}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {61\,A}{160}+\frac {61\,B}{640}\right )\,\sin \left (c+d\,x\right )-\frac {A}{5}+\frac {3\,B}{40}}{d\,\left (-a^2\,{\sin \left (c+d\,x\right )}^8-2\,a^2\,{\sin \left (c+d\,x\right )}^7+2\,a^2\,{\sin \left (c+d\,x\right )}^6+6\,a^2\,{\sin \left (c+d\,x\right )}^5-6\,a^2\,{\sin \left (c+d\,x\right )}^3-2\,a^2\,{\sin \left (c+d\,x\right )}^2+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )}+\frac {7\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (4\,A+B\right )}{128\,a^2\,d} \]
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