Integrand size = 29, antiderivative size = 217 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {187 \log (1-\sin (c+d x))}{256 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {443 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {9 a}{128 d (a-a \sin (c+d x))^2}+\frac {47}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{12 d (a+a \sin (c+d x))^3}-\frac {19 a}{64 d (a+a \sin (c+d x))^2}-\frac {35}{32 d (a+a \sin (c+d x))} \]
-csc(d*x+c)/a/d-187/256*ln(1-sin(d*x+c))/a/d-ln(sin(d*x+c))/a/d+443/256*ln (1+sin(d*x+c))/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3+9/128*a/d/(a-a*sin(d*x+c) )^2+47/128/d/(a-a*sin(d*x+c))-1/64*a^3/d/(a+a*sin(d*x+c))^4-1/12*a^2/d/(a+ a*sin(d*x+c))^3-19/64*a/d/(a+a*sin(d*x+c))^2-35/32/d/(a+a*sin(d*x+c))
Time = 6.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^7 \left (-\frac {\csc (c+d x)}{a^8}-\frac {187 \log (1-\sin (c+d x))}{256 a^8}-\frac {\log (\sin (c+d x))}{a^8}+\frac {443 \log (1+\sin (c+d x))}{256 a^8}+\frac {1}{96 a^5 (a-a \sin (c+d x))^3}+\frac {9}{128 a^6 (a-a \sin (c+d x))^2}+\frac {47}{128 a^7 (a-a \sin (c+d x))}-\frac {1}{64 a^4 (a+a \sin (c+d x))^4}-\frac {1}{12 a^5 (a+a \sin (c+d x))^3}-\frac {19}{64 a^6 (a+a \sin (c+d x))^2}-\frac {35}{32 a^7 (a+a \sin (c+d x))}\right )}{d} \]
(a^7*(-(Csc[c + d*x]/a^8) - (187*Log[1 - Sin[c + d*x]])/(256*a^8) - Log[Si n[c + d*x]]/a^8 + (443*Log[1 + Sin[c + d*x]])/(256*a^8) + 1/(96*a^5*(a - a *Sin[c + d*x])^3) + 9/(128*a^6*(a - a*Sin[c + d*x])^2) + 47/(128*a^7*(a - a*Sin[c + d*x])) - 1/(64*a^4*(a + a*Sin[c + d*x])^4) - 1/(12*a^5*(a + a*Si n[c + d*x])^3) - 19/(64*a^6*(a + a*Sin[c + d*x])^2) - 35/(32*a^7*(a + a*Si n[c + d*x]))))/d
Time = 0.42 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int \frac {\csc ^2(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^9 \int \frac {\csc ^2(c+d x)}{a^2 (a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^9 \int \left (\frac {\csc ^2(c+d x)}{a^{11}}-\frac {\csc (c+d x)}{a^{11}}+\frac {187}{256 a^{10} (a-a \sin (c+d x))}+\frac {443}{256 a^{10} (\sin (c+d x) a+a)}+\frac {47}{128 a^9 (a-a \sin (c+d x))^2}+\frac {35}{32 a^9 (\sin (c+d x) a+a)^2}+\frac {9}{64 a^8 (a-a \sin (c+d x))^3}+\frac {19}{32 a^8 (\sin (c+d x) a+a)^3}+\frac {1}{32 a^7 (a-a \sin (c+d x))^4}+\frac {1}{4 a^7 (\sin (c+d x) a+a)^4}+\frac {1}{16 a^6 (\sin (c+d x) a+a)^5}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^9 \left (-\frac {\csc (c+d x)}{a^{10}}-\frac {\log (a \sin (c+d x))}{a^{10}}-\frac {187 \log (a-a \sin (c+d x))}{256 a^{10}}+\frac {443 \log (a \sin (c+d x)+a)}{256 a^{10}}+\frac {47}{128 a^9 (a-a \sin (c+d x))}-\frac {35}{32 a^9 (a \sin (c+d x)+a)}+\frac {9}{128 a^8 (a-a \sin (c+d x))^2}-\frac {19}{64 a^8 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^7 (a-a \sin (c+d x))^3}-\frac {1}{12 a^7 (a \sin (c+d x)+a)^3}-\frac {1}{64 a^6 (a \sin (c+d x)+a)^4}\right )}{d}\) |
(a^9*(-(Csc[c + d*x]/a^10) - Log[a*Sin[c + d*x]]/a^10 - (187*Log[a - a*Sin [c + d*x]])/(256*a^10) + (443*Log[a + a*Sin[c + d*x]])/(256*a^10) + 1/(96* a^7*(a - a*Sin[c + d*x])^3) + 9/(128*a^8*(a - a*Sin[c + d*x])^2) + 47/(128 *a^9*(a - a*Sin[c + d*x])) - 1/(64*a^6*(a + a*Sin[c + d*x])^4) - 1/(12*a^7 *(a + a*Sin[c + d*x])^3) - 19/(64*a^8*(a + a*Sin[c + d*x])^2) - 35/(32*a^9 *(a + a*Sin[c + d*x]))))/d
3.9.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 1.79 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{\sin \left (d x +c \right )}-\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {47}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {187 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {19}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {35}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {443 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(134\) |
| default | \(\frac {-\frac {1}{\sin \left (d x +c \right )}-\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {9}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {47}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {187 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{12 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {19}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {35}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {443 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(134\) |
| risch | \(-\frac {i \left (1506 i {\mathrm e}^{14 i \left (d x +c \right )}+945 \,{\mathrm e}^{15 i \left (d x +c \right )}+7284 i {\mathrm e}^{12 i \left (d x +c \right )}+4233 \,{\mathrm e}^{13 i \left (d x +c \right )}+12574 i {\mathrm e}^{10 i \left (d x +c \right )}+6549 \,{\mathrm e}^{11 i \left (d x +c \right )}+6424 i {\mathrm e}^{8 i \left (d x +c \right )}+2749 \,{\mathrm e}^{9 i \left (d x +c \right )}+12574 i {\mathrm e}^{6 i \left (d x +c \right )}-2749 \,{\mathrm e}^{7 i \left (d x +c \right )}+7284 i {\mathrm e}^{4 i \left (d x +c \right )}-6549 \,{\mathrm e}^{5 i \left (d x +c \right )}+1506 i {\mathrm e}^{2 i \left (d x +c \right )}-4233 \,{\mathrm e}^{3 i \left (d x +c \right )}-945 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} d a}+\frac {443 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {187 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(287\) |
| parallelrisch | \(\frac {\left (-1122 \cos \left (6 d x +6 c \right )-2805 \sin \left (d x +c \right )-5049 \sin \left (3 d x +3 c \right )-2805 \sin \left (5 d x +5 c \right )-561 \sin \left (7 d x +7 c \right )-16830 \cos \left (2 d x +2 c \right )-6732 \cos \left (4 d x +4 c \right )-11220\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (2658 \cos \left (6 d x +6 c \right )+6645 \sin \left (d x +c \right )+11961 \sin \left (3 d x +3 c \right )+6645 \sin \left (5 d x +5 c \right )+1329 \sin \left (7 d x +7 c \right )+39870 \cos \left (2 d x +2 c \right )+15948 \cos \left (4 d x +4 c \right )+26580\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-768 \cos \left (6 d x +6 c \right )-1920 \sin \left (d x +c \right )-3456 \sin \left (3 d x +3 c \right )-1920 \sin \left (5 d x +5 c \right )-384 \sin \left (7 d x +7 c \right )-11520 \cos \left (2 d x +2 c \right )-4608 \cos \left (4 d x +4 c \right )-7680\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1568 \cos \left (6 d x +6 c \right )-784 \cos \left (7 d x +7 c \right )-111352 \cos \left (d x +c \right )+64704 \cos \left (2 d x +2 c \right )-40068 \cos \left (3 d x +3 c \right )+15432 \cos \left (4 d x +4 c \right )-8500 \cos \left (5 d x +5 c \right )+79000\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-12288 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-322 \cos \left (6 d x +6 c \right )+66 \cos \left (2 d x +2 c \right )-948 \cos \left (4 d x +4 c \right )+1204}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(491\) |
1/d/a*(-1/sin(d*x+c)-ln(sin(d*x+c))-1/96/(sin(d*x+c)-1)^3+9/128/(sin(d*x+c )-1)^2-47/128/(sin(d*x+c)-1)-187/256*ln(sin(d*x+c)-1)-1/64/(1+sin(d*x+c))^ 4-1/12/(1+sin(d*x+c))^3-19/64/(1+sin(d*x+c))^2-35/32/(1+sin(d*x+c))+443/25 6*ln(1+sin(d*x+c)))
Time = 0.30 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.19 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1506 \, \cos \left (d x + c\right )^{6} - 438 \, \cos \left (d x + c\right )^{4} - 188 \, \cos \left (d x + c\right )^{2} - 768 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 1329 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 561 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (945 \, \cos \left (d x + c\right )^{6} - 123 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{6}\right )}} \]
1/768*(1506*cos(d*x + c)^6 - 438*cos(d*x + c)^4 - 188*cos(d*x + c)^2 - 768 *(cos(d*x + c)^8 - cos(d*x + c)^6*sin(d*x + c) - cos(d*x + c)^6)*log(1/2*s in(d*x + c)) + 1329*(cos(d*x + c)^8 - cos(d*x + c)^6*sin(d*x + c) - cos(d* x + c)^6)*log(sin(d*x + c) + 1) - 561*(cos(d*x + c)^8 - cos(d*x + c)^6*sin (d*x + c) - cos(d*x + c)^6)*log(-sin(d*x + c) + 1) + 2*(945*cos(d*x + c)^6 - 123*cos(d*x + c)^4 - 30*cos(d*x + c)^2 - 8)*sin(d*x + c) - 112)/(a*d*co s(d*x + c)^8 - a*d*cos(d*x + c)^6*sin(d*x + c) - a*d*cos(d*x + c)^6)
Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{7} + 753 \, \sin \left (d x + c\right )^{6} - 2712 \, \sin \left (d x + c\right )^{5} - 2040 \, \sin \left (d x + c\right )^{4} + 2559 \, \sin \left (d x + c\right )^{3} + 1727 \, \sin \left (d x + c\right )^{2} - 784 \, \sin \left (d x + c\right ) - 384\right )}}{a \sin \left (d x + c\right )^{8} + a \sin \left (d x + c\right )^{7} - 3 \, a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} + 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right )} - \frac {1329 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {561 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {768 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]
-1/768*(2*(945*sin(d*x + c)^7 + 753*sin(d*x + c)^6 - 2712*sin(d*x + c)^5 - 2040*sin(d*x + c)^4 + 2559*sin(d*x + c)^3 + 1727*sin(d*x + c)^2 - 784*sin (d*x + c) - 384)/(a*sin(d*x + c)^8 + a*sin(d*x + c)^7 - 3*a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 + 3*a*sin(d*x + c)^4 + 3*a*sin(d*x + c)^3 - a*sin(d* x + c)^2 - a*sin(d*x + c)) - 1329*log(sin(d*x + c) + 1)/a + 561*log(sin(d* x + c) - 1)/a + 768*log(sin(d*x + c))/a)/d
Time = 0.45 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5316 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {2244 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {3072 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {3072 \, {\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )} + \frac {2 \, {\left (2057 \, \sin \left (d x + c\right )^{3} - 6735 \, \sin \left (d x + c\right )^{2} + 7407 \, \sin \left (d x + c\right ) - 2745\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {11075 \, \sin \left (d x + c\right )^{4} + 47660 \, \sin \left (d x + c\right )^{3} + 77442 \, \sin \left (d x + c\right )^{2} + 56460 \, \sin \left (d x + c\right ) + 15651}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
1/3072*(5316*log(abs(sin(d*x + c) + 1))/a - 2244*log(abs(sin(d*x + c) - 1) )/a - 3072*log(abs(sin(d*x + c)))/a + 3072*(sin(d*x + c) - 1)/(a*sin(d*x + c)) + 2*(2057*sin(d*x + c)^3 - 6735*sin(d*x + c)^2 + 7407*sin(d*x + c) - 2745)/(a*(sin(d*x + c) - 1)^3) - (11075*sin(d*x + c)^4 + 47660*sin(d*x + c )^3 + 77442*sin(d*x + c)^2 + 56460*sin(d*x + c) + 15651)/(a*(sin(d*x + c) + 1)^4))/d
Time = 10.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.98 \[ \int \frac {\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {443\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {187\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}-\frac {-\frac {315\,{\sin \left (c+d\,x\right )}^7}{128}-\frac {251\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {113\,{\sin \left (c+d\,x\right )}^5}{16}+\frac {85\,{\sin \left (c+d\,x\right )}^4}{16}-\frac {853\,{\sin \left (c+d\,x\right )}^3}{128}-\frac {1727\,{\sin \left (c+d\,x\right )}^2}{384}+\frac {49\,\sin \left (c+d\,x\right )}{24}+1}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^8-a\,{\sin \left (c+d\,x\right )}^7+3\,a\,{\sin \left (c+d\,x\right )}^6+3\,a\,{\sin \left (c+d\,x\right )}^5-3\,a\,{\sin \left (c+d\,x\right )}^4-3\,a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
(443*log(sin(c + d*x) + 1))/(256*a*d) - (187*log(sin(c + d*x) - 1))/(256*a *d) - ((49*sin(c + d*x))/24 - (1727*sin(c + d*x)^2)/384 - (853*sin(c + d*x )^3)/128 + (85*sin(c + d*x)^4)/16 + (113*sin(c + d*x)^5)/16 - (251*sin(c + d*x)^6)/128 - (315*sin(c + d*x)^7)/128 + 1)/(d*(a*sin(c + d*x) + a*sin(c + d*x)^2 - 3*a*sin(c + d*x)^3 - 3*a*sin(c + d*x)^4 + 3*a*sin(c + d*x)^5 + 3*a*sin(c + d*x)^6 - a*sin(c + d*x)^7 - a*sin(c + d*x)^8)) - log(sin(c + d *x))/(a*d)