Integrand size = 29, antiderivative size = 278 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {949 \log (1+\sin (c+d x))}{512 a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}+\frac {5 a^2}{192 d (a-a \sin (c+d x))^3}+\frac {61}{128 d (a-a \sin (c+d x))}-\frac {9 a^3}{256 d (a+a \sin (c+d x))^4}-\frac {47 a^2}{384 d (a+a \sin (c+d x))^3}-\frac {315}{256 d (a+a \sin (c+d x))}-\frac {a^9}{160 d \left (a^2+a^2 \sin (c+d x)\right )^5}+\frac {57 a^9}{512 d \left (a^5-a^5 \sin (c+d x)\right )^2}-\frac {187 a^9}{512 d \left (a^5+a^5 \sin (c+d x)\right )^2} \] Output:
-csc(d*x+c)/a/d-437/512*ln(1-sin(d*x+c))/a/d-ln(sin(d*x+c))/a/d+949/512*ln (1+sin(d*x+c))/a/d+1/256*a^3/d/(a-a*sin(d*x+c))^4+5/192*a^2/d/(a-a*sin(d*x +c))^3+61/128/d/(a-a*sin(d*x+c))-9/256*a^3/d/(a+a*sin(d*x+c))^4-47/384*a^2 /d/(a+a*sin(d*x+c))^3-315/256/d/(a+a*sin(d*x+c))-1/160*a^9/d/(a^2+a^2*sin( d*x+c))^5+57/512*a^9/d/(a^5-a^5*sin(d*x+c))^2-187/512*a^9/d/(a^5+a^5*sin(d *x+c))^2
Time = 6.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.86 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {a^9 \left (-\frac {\csc (c+d x)}{a^{10}}-\frac {437 \log (1-\sin (c+d x))}{512 a^{10}}-\frac {\log (\sin (c+d x))}{a^{10}}+\frac {949 \log (1+\sin (c+d x))}{512 a^{10}}+\frac {1}{256 a^6 (a-a \sin (c+d x))^4}+\frac {5}{192 a^7 (a-a \sin (c+d x))^3}+\frac {57}{512 a^8 (a-a \sin (c+d x))^2}+\frac {61}{128 a^9 (a-a \sin (c+d x))}-\frac {1}{160 a^5 (a+a \sin (c+d x))^5}-\frac {9}{256 a^6 (a+a \sin (c+d x))^4}-\frac {47}{384 a^7 (a+a \sin (c+d x))^3}-\frac {187}{512 a^8 (a+a \sin (c+d x))^2}-\frac {315}{256 a^9 (a+a \sin (c+d x))}\right )}{d} \] Input:
Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]
Output:
(a^9*(-(Csc[c + d*x]/a^10) - (437*Log[1 - Sin[c + d*x]])/(512*a^10) - Log[ Sin[c + d*x]]/a^10 + (949*Log[1 + Sin[c + d*x]])/(512*a^10) + 1/(256*a^6*( a - a*Sin[c + d*x])^4) + 5/(192*a^7*(a - a*Sin[c + d*x])^3) + 57/(512*a^8* (a - a*Sin[c + d*x])^2) + 61/(128*a^9*(a - a*Sin[c + d*x])) - 1/(160*a^5*( a + a*Sin[c + d*x])^5) - 9/(256*a^6*(a + a*Sin[c + d*x])^4) - 47/(384*a^7* (a + a*Sin[c + d*x])^3) - 187/(512*a^8*(a + a*Sin[c + d*x])^2) - 315/(256* a^9*(a + a*Sin[c + d*x]))))/d
Time = 0.51 (sec) , antiderivative size = 245, 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.
| \(\displaystyle \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^2 \cos (c+d x)^9 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^9 \int \frac {\csc ^2(c+d x)}{(a-a \sin (c+d x))^5 (\sin (c+d x) a+a)^6}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^{11} \int \frac {\csc ^2(c+d x)}{a^2 (a-a \sin (c+d x))^5 (\sin (c+d x) a+a)^6}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^{11} \int \left (\frac {\csc ^2(c+d x)}{a^{13}}-\frac {\csc (c+d x)}{a^{13}}+\frac {437}{512 a^{12} (a-a \sin (c+d x))}+\frac {949}{512 a^{12} (\sin (c+d x) a+a)}+\frac {61}{128 a^{11} (a-a \sin (c+d x))^2}+\frac {315}{256 a^{11} (\sin (c+d x) a+a)^2}+\frac {57}{256 a^{10} (a-a \sin (c+d x))^3}+\frac {187}{256 a^{10} (\sin (c+d x) a+a)^3}+\frac {5}{64 a^9 (a-a \sin (c+d x))^4}+\frac {47}{128 a^9 (\sin (c+d x) a+a)^4}+\frac {1}{64 a^8 (a-a \sin (c+d x))^5}+\frac {9}{64 a^8 (\sin (c+d x) a+a)^5}+\frac {1}{32 a^7 (\sin (c+d x) a+a)^6}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{11} \left (-\frac {\csc (c+d x)}{a^{12}}-\frac {\log (a \sin (c+d x))}{a^{12}}-\frac {437 \log (a-a \sin (c+d x))}{512 a^{12}}+\frac {949 \log (a \sin (c+d x)+a)}{512 a^{12}}+\frac {61}{128 a^{11} (a-a \sin (c+d x))}-\frac {315}{256 a^{11} (a \sin (c+d x)+a)}+\frac {57}{512 a^{10} (a-a \sin (c+d x))^2}-\frac {187}{512 a^{10} (a \sin (c+d x)+a)^2}+\frac {5}{192 a^9 (a-a \sin (c+d x))^3}-\frac {47}{384 a^9 (a \sin (c+d x)+a)^3}+\frac {1}{256 a^8 (a-a \sin (c+d x))^4}-\frac {9}{256 a^8 (a \sin (c+d x)+a)^4}-\frac {1}{160 a^7 (a \sin (c+d x)+a)^5}\right )}{d}\) |
Input:
Int[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]
Output:
(a^11*(-(Csc[c + d*x]/a^12) - Log[a*Sin[c + d*x]]/a^12 - (437*Log[a - a*Si n[c + d*x]])/(512*a^12) + (949*Log[a + a*Sin[c + d*x]])/(512*a^12) + 1/(25 6*a^8*(a - a*Sin[c + d*x])^4) + 5/(192*a^9*(a - a*Sin[c + d*x])^3) + 57/(5 12*a^10*(a - a*Sin[c + d*x])^2) + 61/(128*a^11*(a - a*Sin[c + d*x])) - 1/( 160*a^7*(a + a*Sin[c + d*x])^5) - 9/(256*a^8*(a + a*Sin[c + d*x])^4) - 47/ (384*a^9*(a + a*Sin[c + d*x])^3) - 187/(512*a^10*(a + a*Sin[c + d*x])^2) - 315/(256*a^11*(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 = 3.66 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.57
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {9}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {47}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {315}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512}-\frac {1}{\sin \left (d x +c \right )}-\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {5}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {61}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512}}{d a}\) | \(158\) |
| default | \(\frac {-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {9}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {47}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {187}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {315}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512}-\frac {1}{\sin \left (d x +c \right )}-\ln \left (\sin \left (d x +c \right )\right )+\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}-\frac {5}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {57}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {61}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512}}{d a}\) | \(158\) |
| risch | \(-\frac {i \left (431256 i {\mathrm e}^{8 i \left (d x +c \right )}+10395 \,{\mathrm e}^{19 i \left (d x +c \right )}+16950 i {\mathrm e}^{18 i \left (d x +c \right )}+66585 \,{\mathrm e}^{17 i \left (d x +c \right )}+320248 i {\mathrm e}^{14 i \left (d x +c \right )}+170184 \,{\mathrm e}^{15 i \left (d x +c \right )}+320248 i {\mathrm e}^{6 i \left (d x +c \right )}+205264 \,{\mathrm e}^{13 i \left (d x +c \right )}+431256 i {\mathrm e}^{12 i \left (d x +c \right )}+78982 \,{\mathrm e}^{11 i \left (d x +c \right )}+115560 i {\mathrm e}^{16 i \left (d x +c \right )}-78982 \,{\mathrm e}^{9 i \left (d x +c \right )}+16950 i {\mathrm e}^{2 i \left (d x +c \right )}-205264 \,{\mathrm e}^{7 i \left (d x +c \right )}+198052 i {\mathrm e}^{10 i \left (d x +c \right )}-170184 \,{\mathrm e}^{5 i \left (d x +c \right )}+115560 i {\mathrm e}^{4 i \left (d x +c \right )}-66585 \,{\mathrm e}^{3 i \left (d x +c \right )}-10395 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} d a}-\frac {437 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 a d}+\frac {949 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(333\) |
| parallelrisch | \(-\frac {14 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\frac {20729}{53760}+\frac {437 \left (\frac {1}{4}+\sin \left (3 d x +3 c \right )+\frac {\cos \left (2 d x +2 c \right )}{4}+\frac {\sin \left (d x +c \right )}{2}-\frac {\cos \left (10 d x +10 c \right )}{56}+\frac {\sin \left (9 d x +9 c \right )}{28}-\frac {3 \cos \left (8 d x +8 c \right )}{28}+\frac {\sin \left (7 d x +7 c \right )}{4}-\frac {13 \cos \left (6 d x +6 c \right )}{56}-\frac {\cos \left (4 d x +4 c \right )}{7}+\frac {5 \sin \left (5 d x +5 c \right )}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{256}+\frac {949 \left (-\frac {1}{4}-\sin \left (3 d x +3 c \right )-\frac {\cos \left (2 d x +2 c \right )}{4}-\frac {\sin \left (d x +c \right )}{2}+\frac {\cos \left (10 d x +10 c \right )}{56}-\frac {\sin \left (9 d x +9 c \right )}{28}+\frac {3 \cos \left (8 d x +8 c \right )}{28}-\frac {\sin \left (7 d x +7 c \right )}{4}+\frac {13 \cos \left (6 d x +6 c \right )}{56}+\frac {\cos \left (4 d x +4 c \right )}{7}-\frac {5 \sin \left (5 d x +5 c \right )}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{256}+\left (\frac {1}{4}+\sin \left (3 d x +3 c \right )+\frac {\cos \left (2 d x +2 c \right )}{4}+\frac {\sin \left (d x +c \right )}{2}-\frac {\cos \left (10 d x +10 c \right )}{56}+\frac {\sin \left (9 d x +9 c \right )}{28}-\frac {3 \cos \left (8 d x +8 c \right )}{28}+\frac {\sin \left (7 d x +7 c \right )}{4}-\frac {13 \cos \left (6 d x +6 c \right )}{56}-\frac {\cos \left (4 d x +4 c \right )}{7}+\frac {5 \sin \left (5 d x +5 c \right )}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {6673 \cos \left (2 d x +2 c \right )}{1920}+\frac {44143 \cos \left (4 d x +4 c \right )}{13440}+\frac {21127 \cos \left (6 d x +6 c \right )}{13440}+\frac {257 \cos \left (10 d x +10 c \right )}{6720}+\frac {9017 \sin \left (7 d x +7 c \right )}{107520}+\frac {2171 \sin \left (9 d x +9 c \right )}{107520}+\frac {991 \cos \left (8 d x +8 c \right )}{2560}-\frac {18077 \sin \left (d x +c \right )}{53760}-\frac {521 \sin \left (3 d x +3 c \right )}{2240}+\frac {713 \sin \left (5 d x +5 c \right )}{13440}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (\sin \left (9 d x +9 c \right )+7 \sin \left (7 d x +7 c \right )+20 \sin \left (5 d x +5 c \right )+28 \sin \left (3 d x +3 c \right )+14 \sin \left (d x +c \right )+2 \cos \left (8 d x +8 c \right )+16 \cos \left (6 d x +6 c \right )+56 \cos \left (4 d x +4 c \right )+112 \cos \left (2 d x +2 c \right )+70\right )}\) | \(596\) |
Input:
int(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-1/160/(1+sin(d*x+c))^5-9/256/(1+sin(d*x+c))^4-47/384/(1+sin(d*x+c) )^3-187/512/(1+sin(d*x+c))^2-315/256/(1+sin(d*x+c))+949/512*ln(1+sin(d*x+c ))-1/sin(d*x+c)-ln(sin(d*x+c))+1/256/(sin(d*x+c)-1)^4-5/192/(sin(d*x+c)-1) ^3+57/512/(sin(d*x+c)-1)^2-61/128/(sin(d*x+c)-1)-437/512*ln(sin(d*x+c)-1))
Time = 0.12 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {16950 \, \cos \left (d x + c\right )^{8} - 5010 \, \cos \left (d x + c\right )^{6} - 2132 \, \cos \left (d x + c\right )^{4} - 1264 \, \cos \left (d x + c\right )^{2} - 7680 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 14235 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6555 \, {\left (\cos \left (d x + c\right )^{10} - \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (10395 \, \cos \left (d x + c\right )^{8} - 1545 \, \cos \left (d x + c\right )^{6} - 426 \, \cos \left (d x + c\right )^{4} - 152 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) - 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{10} - a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{8}\right )}} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/7680*(16950*cos(d*x + c)^8 - 5010*cos(d*x + c)^6 - 2132*cos(d*x + c)^4 - 1264*cos(d*x + c)^2 - 7680*(cos(d*x + c)^10 - cos(d*x + c)^8*sin(d*x + c) - cos(d*x + c)^8)*log(1/2*sin(d*x + c)) + 14235*(cos(d*x + c)^10 - cos(d* x + c)^8*sin(d*x + c) - cos(d*x + c)^8)*log(sin(d*x + c) + 1) - 6555*(cos( d*x + c)^10 - cos(d*x + c)^8*sin(d*x + c) - cos(d*x + c)^8)*log(-sin(d*x + c) + 1) + 2*(10395*cos(d*x + c)^8 - 1545*cos(d*x + c)^6 - 426*cos(d*x + c )^4 - 152*cos(d*x + c)^2 - 48)*sin(d*x + c) - 864)/(a*d*cos(d*x + c)^10 - a*d*cos(d*x + c)^8*sin(d*x + c) - a*d*cos(d*x + c)^8)
Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**2*sec(d*x+c)**9/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (10395 \, \sin \left (d x + c\right )^{9} + 8475 \, \sin \left (d x + c\right )^{8} - 40035 \, \sin \left (d x + c\right )^{7} - 31395 \, \sin \left (d x + c\right )^{6} + 57309 \, \sin \left (d x + c\right )^{5} + 42269 \, \sin \left (d x + c\right )^{4} - 35941 \, \sin \left (d x + c\right )^{3} - 23621 \, \sin \left (d x + c\right )^{2} + 8224 \, \sin \left (d x + c\right ) + 3840\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 4 \, a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} + 6 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 4 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right )} - \frac {14235 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {6555 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {7680 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{7680 \, d} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/7680*(2*(10395*sin(d*x + c)^9 + 8475*sin(d*x + c)^8 - 40035*sin(d*x + c )^7 - 31395*sin(d*x + c)^6 + 57309*sin(d*x + c)^5 + 42269*sin(d*x + c)^4 - 35941*sin(d*x + c)^3 - 23621*sin(d*x + c)^2 + 8224*sin(d*x + c) + 3840)/( a*sin(d*x + c)^10 + a*sin(d*x + c)^9 - 4*a*sin(d*x + c)^8 - 4*a*sin(d*x + c)^7 + 6*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 - 4*a*sin(d*x + c)^4 - 4*a* sin(d*x + c)^3 + a*sin(d*x + c)^2 + a*sin(d*x + c)) - 14235*log(sin(d*x + c) + 1)/a + 6555*log(sin(d*x + c) - 1)/a + 7680*log(sin(d*x + c))/a)/d
Time = 0.15 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.64 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {949 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{512 \, a d} - \frac {437 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{512 \, a d} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a d} - \frac {10395 \, \sin \left (d x + c\right )^{9} + 8475 \, \sin \left (d x + c\right )^{8} - 40035 \, \sin \left (d x + c\right )^{7} - 31395 \, \sin \left (d x + c\right )^{6} + 57309 \, \sin \left (d x + c\right )^{5} + 42269 \, \sin \left (d x + c\right )^{4} - 35941 \, \sin \left (d x + c\right )^{3} - 23621 \, \sin \left (d x + c\right )^{2} + 8224 \, \sin \left (d x + c\right ) + 3840}{3840 \, a d {\left (\sin \left (d x + c\right ) + 1\right )}^{5} {\left (\sin \left (d x + c\right ) - 1\right )}^{4} \sin \left (d x + c\right )} \] Input:
integrate(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
949/512*log(abs(sin(d*x + c) + 1))/(a*d) - 437/512*log(abs(sin(d*x + c) - 1))/(a*d) - log(abs(sin(d*x + c)))/(a*d) - 1/3840*(10395*sin(d*x + c)^9 + 8475*sin(d*x + c)^8 - 40035*sin(d*x + c)^7 - 31395*sin(d*x + c)^6 + 57309* sin(d*x + c)^5 + 42269*sin(d*x + c)^4 - 35941*sin(d*x + c)^3 - 23621*sin(d *x + c)^2 + 8224*sin(d*x + c) + 3840)/(a*d*(sin(d*x + c) + 1)^5*(sin(d*x + c) - 1)^4*sin(d*x + c))
Time = 32.45 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.91 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {949\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{512\,a\,d}-\frac {437\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{512\,a\,d}-\frac {\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d}-\frac {\frac {693\,{\sin \left (c+d\,x\right )}^9}{256}+\frac {565\,{\sin \left (c+d\,x\right )}^8}{256}-\frac {2669\,{\sin \left (c+d\,x\right )}^7}{256}-\frac {2093\,{\sin \left (c+d\,x\right )}^6}{256}+\frac {19103\,{\sin \left (c+d\,x\right )}^5}{1280}+\frac {42269\,{\sin \left (c+d\,x\right )}^4}{3840}-\frac {35941\,{\sin \left (c+d\,x\right )}^3}{3840}-\frac {23621\,{\sin \left (c+d\,x\right )}^2}{3840}+\frac {257\,\sin \left (c+d\,x\right )}{120}+1}{d\,\left (a\,{\sin \left (c+d\,x\right )}^{10}+a\,{\sin \left (c+d\,x\right )}^9-4\,a\,{\sin \left (c+d\,x\right )}^8-4\,a\,{\sin \left (c+d\,x\right )}^7+6\,a\,{\sin \left (c+d\,x\right )}^6+6\,a\,{\sin \left (c+d\,x\right )}^5-4\,a\,{\sin \left (c+d\,x\right )}^4-4\,a\,{\sin \left (c+d\,x\right )}^3+a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )\right )} \] Input:
int(1/(cos(c + d*x)^9*sin(c + d*x)^2*(a + a*sin(c + d*x))),x)
Output:
(949*log(sin(c + d*x) + 1))/(512*a*d) - (437*log(sin(c + d*x) - 1))/(512*a *d) - log(sin(c + d*x))/(a*d) - ((257*sin(c + d*x))/120 - (23621*sin(c + d *x)^2)/3840 - (35941*sin(c + d*x)^3)/3840 + (42269*sin(c + d*x)^4)/3840 + (19103*sin(c + d*x)^5)/1280 - (2093*sin(c + d*x)^6)/256 - (2669*sin(c + d* x)^7)/256 + (565*sin(c + d*x)^8)/256 + (693*sin(c + d*x)^9)/256 + 1)/(d*(a *sin(c + d*x) + a*sin(c + d*x)^2 - 4*a*sin(c + d*x)^3 - 4*a*sin(c + d*x)^4 + 6*a*sin(c + d*x)^5 + 6*a*sin(c + d*x)^6 - 4*a*sin(c + d*x)^7 - 4*a*sin( c + d*x)^8 + a*sin(c + d*x)^9 + a*sin(c + d*x)^10))
Time = 0.20 (sec) , antiderivative size = 836, normalized size of antiderivative = 3.01 \[ \int \frac {\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^2*sec(d*x+c)^9/(a+a*sin(d*x+c)),x)
Output:
( - 6555*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**10 - 6555*log(tan((c + d* x)/2) - 1)*sin(c + d*x)**9 + 26220*log(tan((c + d*x)/2) - 1)*sin(c + d*x)* *8 + 26220*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7 - 39330*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6 - 39330*log(tan((c + d*x)/2) - 1)*sin(c + d*x )**5 + 26220*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 26220*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**3 - 6555*log(tan((c + d*x)/2) - 1)*sin(c + d* x)**2 - 6555*log(tan((c + d*x)/2) - 1)*sin(c + d*x) + 14235*log(tan((c + d *x)/2) + 1)*sin(c + d*x)**10 + 14235*log(tan((c + d*x)/2) + 1)*sin(c + d*x )**9 - 56940*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8 - 56940*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**7 + 85410*log(tan((c + d*x)/2) + 1)*sin(c + d *x)**6 + 85410*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**5 - 56940*log(tan(( c + d*x)/2) + 1)*sin(c + d*x)**4 - 56940*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 + 14235*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 14235*log(tan ((c + d*x)/2) + 1)*sin(c + d*x) - 3840*log(tan((c + d*x)/2))*sin(c + d*x)* *10 - 3840*log(tan((c + d*x)/2))*sin(c + d*x)**9 + 15360*log(tan((c + d*x) /2))*sin(c + d*x)**8 + 15360*log(tan((c + d*x)/2))*sin(c + d*x)**7 - 23040 *log(tan((c + d*x)/2))*sin(c + d*x)**6 - 23040*log(tan((c + d*x)/2))*sin(c + d*x)**5 + 15360*log(tan((c + d*x)/2))*sin(c + d*x)**4 + 15360*log(tan(( c + d*x)/2))*sin(c + d*x)**3 - 3840*log(tan((c + d*x)/2))*sin(c + d*x)**2 - 3840*log(tan((c + d*x)/2))*sin(c + d*x) - 14021*sin(c + d*x)**10 - 24...