Integrand size = 29, antiderivative size = 248 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {5 \log (\sin (c+d x))}{a d}-\frac {955 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {69}{128 d (a-a \sin (c+d x))}+\frac {5 a^2}{48 d (a+a \sin (c+d x))^3}+\frac {2}{d (a+a \sin (c+d x))}+\frac {a^7}{64 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {11 a^7}{128 d \left (a^4-a^4 \sin (c+d x)\right )^2}+\frac {29 a^7}{64 d \left (a^4+a^4 \sin (c+d x)\right )^2} \] Output:
csc(d*x+c)/a/d-1/2*csc(d*x+c)^2/a/d-325/256*ln(1-sin(d*x+c))/a/d+5*ln(sin( d*x+c))/a/d-955/256*ln(1+sin(d*x+c))/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3+69/ 128/d/(a-a*sin(d*x+c))+5/48*a^2/d/(a+a*sin(d*x+c))^3+2/d/(a+a*sin(d*x+c))+ 1/64*a^7/d/(a^2+a^2*sin(d*x+c))^4+11/128*a^7/d/(a^4-a^4*sin(d*x+c))^2+29/6 4*a^7/d/(a^4+a^4*sin(d*x+c))^2
Time = 6.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.86 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^7 \left (\frac {\csc (c+d x)}{a^8}-\frac {\csc ^2(c+d x)}{2 a^8}-\frac {325 \log (1-\sin (c+d x))}{256 a^8}+\frac {5 \log (\sin (c+d x))}{a^8}-\frac {955 \log (1+\sin (c+d x))}{256 a^8}+\frac {1}{96 a^5 (a-a \sin (c+d x))^3}+\frac {11}{128 a^6 (a-a \sin (c+d x))^2}+\frac {69}{128 a^7 (a-a \sin (c+d x))}+\frac {1}{64 a^4 (a+a \sin (c+d x))^4}+\frac {5}{48 a^5 (a+a \sin (c+d x))^3}+\frac {29}{64 a^6 (a+a \sin (c+d x))^2}+\frac {2}{a^7 (a+a \sin (c+d x))}\right )}{d} \] Input:
Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]
Output:
(a^7*(Csc[c + d*x]/a^8 - Csc[c + d*x]^2/(2*a^8) - (325*Log[1 - Sin[c + d*x ]])/(256*a^8) + (5*Log[Sin[c + d*x]])/a^8 - (955*Log[1 + Sin[c + d*x]])/(2 56*a^8) + 1/(96*a^5*(a - a*Sin[c + d*x])^3) + 11/(128*a^6*(a - a*Sin[c + d *x])^2) + 69/(128*a^7*(a - a*Sin[c + d*x])) + 1/(64*a^4*(a + a*Sin[c + d*x ])^4) + 5/(48*a^5*(a + a*Sin[c + d*x])^3) + 29/(64*a^6*(a + a*Sin[c + d*x] )^2) + 2/(a^7*(a + a*Sin[c + d*x]))))/d
Time = 0.47 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int \frac {\csc ^3(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^{10} \int \frac {\csc ^3(c+d x)}{a^3 (a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^{10} \int \left (\frac {\csc ^3(c+d x)}{a^{12}}-\frac {\csc ^2(c+d x)}{a^{12}}+\frac {5 \csc (c+d x)}{a^{12}}+\frac {325}{256 a^{11} (a-a \sin (c+d x))}-\frac {955}{256 a^{11} (\sin (c+d x) a+a)}+\frac {69}{128 a^{10} (a-a \sin (c+d x))^2}-\frac {2}{a^{10} (\sin (c+d x) a+a)^2}+\frac {11}{64 a^9 (a-a \sin (c+d x))^3}-\frac {29}{32 a^9 (\sin (c+d x) a+a)^3}+\frac {1}{32 a^8 (a-a \sin (c+d x))^4}-\frac {5}{16 a^8 (\sin (c+d x) a+a)^4}-\frac {1}{16 a^7 (\sin (c+d x) a+a)^5}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{10} \left (-\frac {\csc ^2(c+d x)}{2 a^{11}}+\frac {\csc (c+d x)}{a^{11}}+\frac {5 \log (a \sin (c+d x))}{a^{11}}-\frac {325 \log (a-a \sin (c+d x))}{256 a^{11}}-\frac {955 \log (a \sin (c+d x)+a)}{256 a^{11}}+\frac {69}{128 a^{10} (a-a \sin (c+d x))}+\frac {2}{a^{10} (a \sin (c+d x)+a)}+\frac {11}{128 a^9 (a-a \sin (c+d x))^2}+\frac {29}{64 a^9 (a \sin (c+d x)+a)^2}+\frac {1}{96 a^8 (a-a \sin (c+d x))^3}+\frac {5}{48 a^8 (a \sin (c+d x)+a)^3}+\frac {1}{64 a^7 (a \sin (c+d x)+a)^4}\right )}{d}\) |
Input:
Int[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]
Output:
(a^10*(Csc[c + d*x]/a^11 - Csc[c + d*x]^2/(2*a^11) + (5*Log[a*Sin[c + d*x] ])/a^11 - (325*Log[a - a*Sin[c + d*x]])/(256*a^11) - (955*Log[a + a*Sin[c + d*x]])/(256*a^11) + 1/(96*a^8*(a - a*Sin[c + d*x])^3) + 11/(128*a^9*(a - a*Sin[c + d*x])^2) + 69/(128*a^10*(a - a*Sin[c + d*x])) + 1/(64*a^7*(a + a*Sin[c + d*x])^4) + 5/(48*a^8*(a + a*Sin[c + d*x])^3) + 29/(64*a^9*(a + a *Sin[c + d*x])^2) + 2/(a^10*(a + a*Sin[c + d*x]))))/d
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 = 2.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.57
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {29}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {2}{1+\sin \left (d x +c \right )}-\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {11}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {69}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}}{d a}\) | \(142\) |
| default | \(\frac {\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {5}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {29}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {2}{1+\sin \left (d x +c \right )}-\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {1}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {11}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {69}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}}{d a}\) | \(142\) |
| risch | \(\frac {i \left (14604 \,{\mathrm e}^{13 i \left (d x +c \right )}+945 \,{\mathrm e}^{i \left (d x +c \right )}+9512 \,{\mathrm e}^{11 i \left (d x +c \right )}+6360 \,{\mathrm e}^{15 i \left (d x +c \right )}+945 \,{\mathrm e}^{17 i \left (d x +c \right )}-13690 \,{\mathrm e}^{9 i \left (d x +c \right )}-4602 i {\mathrm e}^{8 i \left (d x +c \right )}+6360 \,{\mathrm e}^{3 i \left (d x +c \right )}+1170 i {\mathrm e}^{14 i \left (d x +c \right )}-4778 i {\mathrm e}^{6 i \left (d x +c \right )}+4778 i {\mathrm e}^{12 i \left (d x +c \right )}-30 i {\mathrm e}^{16 i \left (d x +c \right )}+30 i {\mathrm e}^{2 i \left (d x +c \right )}+4602 i {\mathrm e}^{10 i \left (d x +c \right )}-1170 i {\mathrm e}^{4 i \left (d x +c \right )}+9512 \,{\mathrm e}^{7 i \left (d x +c \right )}+14604 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {955 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {325 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 a d}+\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(310\) |
| parallelrisch | \(\frac {5 \left (-\frac {391}{1536}+\frac {65 \left (-\frac {5}{4}+\frac {\cos \left (8 d x +8 c \right )}{4}+\cos \left (6 d x +6 c \right )+\frac {3 \sin \left (7 d x +7 c \right )}{8}-\cos \left (2 d x +2 c \right )-\frac {3 \sin \left (d x +c \right )}{4}-\sin \left (3 d x +3 c \right )+\cos \left (4 d x +4 c \right )+\frac {\sin \left (9 d x +9 c \right )}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128}+\frac {191 \left (-\frac {5}{4}+\frac {\cos \left (8 d x +8 c \right )}{4}+\cos \left (6 d x +6 c \right )+\frac {3 \sin \left (7 d x +7 c \right )}{8}-\cos \left (2 d x +2 c \right )-\frac {3 \sin \left (d x +c \right )}{4}-\sin \left (3 d x +3 c \right )+\cos \left (4 d x +4 c \right )+\frac {\sin \left (9 d x +9 c \right )}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128}+\left (-\frac {\cos \left (8 d x +8 c \right )}{4}-\cos \left (6 d x +6 c \right )-\frac {3 \sin \left (7 d x +7 c \right )}{8}+\cos \left (2 d x +2 c \right )+\frac {5}{4}+\frac {3 \sin \left (d x +c \right )}{4}+\sin \left (3 d x +3 c \right )-\cos \left (4 d x +4 c \right )-\frac {\sin \left (9 d x +9 c \right )}{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2069 \cos \left (2 d x +2 c \right )}{960}-\frac {1891 \cos \left (4 d x +4 c \right )}{1920}+\frac {17 \cos \left (6 d x +6 c \right )}{192}+\frac {87 \sin \left (7 d x +7 c \right )}{256}+\frac {11 \sin \left (9 d x +9 c \right )}{96}+\frac {163 \cos \left (8 d x +8 c \right )}{1536}-\frac {113 \sin \left (d x +c \right )}{1280}-\frac {377 \sin \left (3 d x +3 c \right )}{1280}+\frac {39 \sin \left (5 d x +5 c \right )}{256}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 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 )}\) | \(493\) |
Input:
int(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(1/64/(1+sin(d*x+c))^4+5/48/(1+sin(d*x+c))^3+29/64/(1+sin(d*x+c))^2+ 2/(1+sin(d*x+c))-955/256*ln(1+sin(d*x+c))-1/2/sin(d*x+c)^2+1/sin(d*x+c)+5* ln(sin(d*x+c))-1/96/(sin(d*x+c)-1)^3+11/128/(sin(d*x+c)-1)^2-69/128/(sin(d *x+c)-1)-325/256*ln(sin(d*x+c)-1))
Time = 0.10 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.25 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1890 \, \cos \left (d x + c\right )^{8} - 600 \, \cos \left (d x + c\right )^{6} - 582 \, \cos \left (d x + c\right )^{4} - 212 \, \cos \left (d x + c\right )^{2} + 3840 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2865 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 975 \, {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6} + {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 165 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6} + {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/768*(1890*cos(d*x + c)^8 - 600*cos(d*x + c)^6 - 582*cos(d*x + c)^4 - 212 *cos(d*x + c)^2 + 3840*(cos(d*x + c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(1/2*sin(d*x + c)) - 2865*(cos(d*x + c) ^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log( sin(d*x + c) + 1) - 975*(cos(d*x + c)^8 - cos(d*x + c)^6 + (cos(d*x + c)^8 - cos(d*x + c)^6)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(15*cos(d*x + c)^6 - 165*cos(d*x + c)^4 - 34*cos(d*x + c)^2 - 8)*sin(d*x + c) - 112)/(a* d*cos(d*x + c)^8 - a*d*cos(d*x + c)^6 + (a*d*cos(d*x + c)^8 - a*d*cos(d*x + c)^6)*sin(d*x + c))
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sec(d*x+c)**7/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (945 \, \sin \left (d x + c\right )^{8} - 15 \, \sin \left (d x + c\right )^{7} - 3480 \, \sin \left (d x + c\right )^{6} - 120 \, \sin \left (d x + c\right )^{5} + 4479 \, \sin \left (d x + c\right )^{4} + 319 \, \sin \left (d x + c\right )^{3} - 2192 \, \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right ) + 192\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 3 \, a \sin \left (d x + c\right )^{7} - 3 \, a \sin \left (d x + c\right )^{6} + 3 \, a \sin \left (d x + c\right )^{5} + 3 \, a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3} - a \sin \left (d x + c\right )^{2}} - \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {3840 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/768*(2*(945*sin(d*x + c)^8 - 15*sin(d*x + c)^7 - 3480*sin(d*x + c)^6 - 1 20*sin(d*x + c)^5 + 4479*sin(d*x + c)^4 + 319*sin(d*x + c)^3 - 2192*sin(d* x + c)^2 - 192*sin(d*x + c) + 192)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8 - 3*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 - a*sin(d*x + c)^3 - a*sin(d*x + c)^2) - 2865*log(sin(d*x + c) + 1 )/a - 975*log(sin(d*x + c) - 1)/a + 3840*log(sin(d*x + c))/a)/d
Time = 0.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {955 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{256 \, a d} - \frac {325 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{256 \, a d} + \frac {5 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a d} + \frac {945 \, \sin \left (d x + c\right )^{8} - 15 \, \sin \left (d x + c\right )^{7} - 3480 \, \sin \left (d x + c\right )^{6} - 120 \, \sin \left (d x + c\right )^{5} + 4479 \, \sin \left (d x + c\right )^{4} + 319 \, \sin \left (d x + c\right )^{3} - 2192 \, \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right ) + 192}{384 \, a d {\left (\sin \left (d x + c\right ) + 1\right )}^{4} {\left (\sin \left (d x + c\right ) - 1\right )}^{3} \sin \left (d x + c\right )^{2}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-955/256*log(abs(sin(d*x + c) + 1))/(a*d) - 325/256*log(abs(sin(d*x + c) - 1))/(a*d) + 5*log(abs(sin(d*x + c)))/(a*d) + 1/384*(945*sin(d*x + c)^8 - 15*sin(d*x + c)^7 - 3480*sin(d*x + c)^6 - 120*sin(d*x + c)^5 + 4479*sin(d* x + c)^4 + 319*sin(d*x + c)^3 - 2192*sin(d*x + c)^2 - 192*sin(d*x + c) + 1 92)/(a*d*(sin(d*x + c) + 1)^4*(sin(d*x + c) - 1)^3*sin(d*x + c)^2)
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.90 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {955\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {325\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {-\frac {315\,{\sin \left (c+d\,x\right )}^8}{128}+\frac {5\,{\sin \left (c+d\,x\right )}^7}{128}+\frac {145\,{\sin \left (c+d\,x\right )}^6}{16}+\frac {5\,{\sin \left (c+d\,x\right )}^5}{16}-\frac {1493\,{\sin \left (c+d\,x\right )}^4}{128}-\frac {319\,{\sin \left (c+d\,x\right )}^3}{384}+\frac {137\,{\sin \left (c+d\,x\right )}^2}{24}+\frac {\sin \left (c+d\,x\right )}{2}-\frac {1}{2}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^9-a\,{\sin \left (c+d\,x\right )}^8+3\,a\,{\sin \left (c+d\,x\right )}^7+3\,a\,{\sin \left (c+d\,x\right )}^6-3\,a\,{\sin \left (c+d\,x\right )}^5-3\,a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2\right )} \] Input:
int(1/(cos(c + d*x)^7*sin(c + d*x)^3*(a + a*sin(c + d*x))),x)
Output:
(5*log(sin(c + d*x)))/(a*d) - (955*log(sin(c + d*x) + 1))/(256*a*d) - (325 *log(sin(c + d*x) - 1))/(256*a*d) + (sin(c + d*x)/2 + (137*sin(c + d*x)^2) /24 - (319*sin(c + d*x)^3)/384 - (1493*sin(c + d*x)^4)/128 + (5*sin(c + d* x)^5)/16 + (145*sin(c + d*x)^6)/16 + (5*sin(c + d*x)^7)/128 - (315*sin(c + d*x)^8)/128 - 1/2)/(d*(a*sin(c + d*x)^2 + 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 + 3*a*sin(c + d*x)^7 - a* sin(c + d*x)^8 - a*sin(c + d*x)^9))
Time = 0.28 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.77 \[ \int \frac {\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^3*sec(d*x+c)^7/(a+a*sin(d*x+c)),x)
Output:
( - 975*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**9 - 975*log(tan((c + d*x)/ 2) - 1)*sin(c + d*x)**8 + 2925*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7 + 2925*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6 - 2925*log(tan((c + d*x)/2 ) - 1)*sin(c + d*x)**5 - 2925*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 975*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**3 + 975*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 - 2865*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**9 - 286 5*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8 + 8595*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**7 + 8595*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6 - 8595 *log(tan((c + d*x)/2) + 1)*sin(c + d*x)**5 - 8595*log(tan((c + d*x)/2) + 1 )*sin(c + d*x)**4 + 2865*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 + 2865* log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 1920*log(tan((c + d*x)/2))*sin (c + d*x)**9 + 1920*log(tan((c + d*x)/2))*sin(c + d*x)**8 - 5760*log(tan(( c + d*x)/2))*sin(c + d*x)**7 - 5760*log(tan((c + d*x)/2))*sin(c + d*x)**6 + 5760*log(tan((c + d*x)/2))*sin(c + d*x)**5 + 5760*log(tan((c + d*x)/2))* sin(c + d*x)**4 - 1920*log(tan((c + d*x)/2))*sin(c + d*x)**3 - 1920*log(ta n((c + d*x)/2))*sin(c + d*x)**2 - 65*sin(c + d*x)**9 + 880*sin(c + d*x)**8 + 180*sin(c + d*x)**7 - 3285*sin(c + d*x)**6 - 315*sin(c + d*x)**5 + 4284 *sin(c + d*x)**4 + 384*sin(c + d*x)**3 - 2127*sin(c + d*x)**2 - 192*sin(c + d*x) + 192)/(384*sin(c + d*x)**2*a*d*(sin(c + d*x)**7 + sin(c + d*x)**6 - 3*sin(c + d*x)**5 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**3 + 3*sin(c +...