Integrand size = 29, antiderivative size = 149 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 \text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {6 \tan (c+d x)}{a^2 d}-\frac {2 \tan ^3(c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d} \] Output:
-9/2*arctanh(cos(d*x+c))/a^2/d+2*cot(d*x+c)/a^2/d-1/2*cot(d*x+c)*csc(d*x+c )/a^2/d+4*sec(d*x+c)/a^2/d+sec(d*x+c)^3/a^2/d+2/5*sec(d*x+c)^5/a^2/d-6*tan (d*x+c)/a^2/d-2*tan(d*x+c)^3/a^2/d-2/5*tan(d*x+c)^5/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(328\) vs. \(2(149)=298\).
Time = 2.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.20 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^2(c+d x) \sec (c+d x) \left (-348+176 \cos (2 (c+d x))-651 \cos (3 (c+d x))+332 \cos (4 (c+d x))+93 \cos (5 (c+d x))-630 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+90 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+18 \cos (c+d x) \left (31+30 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-30 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+630 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-90 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-432 \sin (c+d x)+744 \sin (2 (c+d x))+720 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-720 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (2 (c+d x))-176 \sin (3 (c+d x))-372 \sin (4 (c+d x))-360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+128 \sin (5 (c+d x))\right )}{320 a^2 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
-1/320*(Csc[c + d*x]^2*Sec[c + d*x]*(-348 + 176*Cos[2*(c + d*x)] - 651*Cos [3*(c + d*x)] + 332*Cos[4*(c + d*x)] + 93*Cos[5*(c + d*x)] - 630*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 90*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 18*Cos[c + d*x]*(31 + 30*Log[Cos[(c + d*x)/2]] - 30*Log[Sin[(c + d*x)/2] ]) + 630*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 90*Cos[5*(c + d*x)]*Log[ Sin[(c + d*x)/2]] - 432*Sin[c + d*x] + 744*Sin[2*(c + d*x)] + 720*Log[Cos[ (c + d*x)/2]]*Sin[2*(c + d*x)] - 720*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x) ] - 176*Sin[3*(c + d*x)] - 372*Sin[4*(c + d*x)] - 360*Log[Cos[(c + d*x)/2] ]*Sin[4*(c + d*x)] + 360*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 128*Sin[ 5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)
Time = 0.63 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3352, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \cos (c+d x)^2 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \csc ^3(c+d x) \sec ^6(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^2}{\cos (c+d x)^6 \sin (c+d x)^3}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \csc ^3(c+d x) \sec ^6(c+d x)-2 a^2 \csc ^2(c+d x) \sec ^6(c+d x)+a^2 \csc (c+d x) \sec ^6(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a^2 \tan ^5(c+d x)}{5 d}-\frac {2 a^2 \tan ^3(c+d x)}{d}-\frac {6 a^2 \tan (c+d x)}{d}+\frac {2 a^2 \cot (c+d x)}{d}+\frac {9 a^2 \sec ^5(c+d x)}{10 d}+\frac {3 a^2 \sec ^3(c+d x)}{2 d}+\frac {9 a^2 \sec (c+d x)}{2 d}-\frac {a^2 \csc ^2(c+d x) \sec ^5(c+d x)}{2 d}}{a^4}\) |
Input:
Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
((-9*a^2*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a^2*Cot[c + d*x])/d + (9*a^2*Se c[c + d*x])/(2*d) + (3*a^2*Sec[c + d*x]^3)/(2*d) + (9*a^2*Sec[c + d*x]^5)/ (10*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x]^5)/(2*d) - (6*a^2*Tan[c + d*x])/ d - (2*a^2*Tan[c + d*x]^3)/d - (2*a^2*Tan[c + d*x]^5)/(5*d))/a^4
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 1.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+18 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {20}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {22}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{4 d \,a^{2}}\) | \(162\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+18 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {20}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {22}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{4 d \,a^{2}}\) | \(162\) |
| parallelrisch | \(\frac {180 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+860 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2005 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+1000 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+5 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1505 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-20 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-1948 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-702}{40 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(186\) |
| risch | \(\frac {180 i {\mathrm e}^{8 i \left (d x +c \right )}+45 \,{\mathrm e}^{9 i \left (d x +c \right )}-300 i {\mathrm e}^{6 i \left (d x +c \right )}-300 \,{\mathrm e}^{7 i \left (d x +c \right )}-84 i {\mathrm e}^{4 i \left (d x +c \right )}+174 \,{\mathrm e}^{5 i \left (d x +c \right )}+268 i {\mathrm e}^{2 i \left (d x +c \right )}+212 \,{\mathrm e}^{3 i \left (d x +c \right )}-64 i-211 \,{\mathrm e}^{i \left (d x +c \right )}}{5 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d \,a^{2}}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}\) | \(197\) |
| norman | \(\frac {\frac {1}{8 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2 d a}-\frac {43 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d a}-\frac {43 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d a}+\frac {93 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 d a}-\frac {136 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{5 d a}-\frac {487 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{40 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {9 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(239\) |
Input:
int(csc(d*x+c)^3*sec(d*x+c)^2/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/4/d/a^2*(1/2*tan(1/2*d*x+1/2*c)^2-4*tan(1/2*d*x+1/2*c)-1/2/tan(1/2*d*x+1 /2*c)^2+4/tan(1/2*d*x+1/2*c)+18*ln(tan(1/2*d*x+1/2*c))+16/5/(tan(1/2*d*x+1 /2*c)+1)^5-8/(tan(1/2*d*x+1/2*c)+1)^4+20/(tan(1/2*d*x+1/2*c)+1)^3-22/(tan( 1/2*d*x+1/2*c)+1)^2+49/(tan(1/2*d*x+1/2*c)+1)-1/(tan(1/2*d*x+1/2*c)-1))
Time = 0.12 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {166 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} + 45 \, {\left (\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{3} - 2 \, {\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (32 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 12}{20 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 3 \, a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right ) - 2 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/20*(166*cos(d*x + c)^4 - 144*cos(d*x + c)^2 + 45*(cos(d*x + c)^5 - 3*co s(d*x + c)^3 - 2*(cos(d*x + c)^3 - cos(d*x + c))*sin(d*x + c) + 2*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 45*(cos(d*x + c)^5 - 3*cos(d*x + c)^3 - 2*(cos(d*x + c)^3 - cos(d*x + c))*sin(d*x + c) + 2*cos(d*x + c))*log(-1/ 2*cos(d*x + c) + 1/2) + 4*(32*cos(d*x + c)^4 - 35*cos(d*x + c)^2 - 2)*sin( d*x + c) - 12)/(a^2*d*cos(d*x + c)^5 - 3*a^2*d*cos(d*x + c)^3 + 2*a^2*d*co s(d*x + c) - 2*(a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sec(d*x+c)**2/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (141) = 282\).
Time = 0.04 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.38 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {20 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {567 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1448 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {985 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {820 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1355 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {520 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 5}{\frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {5 \, {\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{2}} + \frac {180 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{40 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/40*((20*sin(d*x + c)/(cos(d*x + c) + 1) + 567*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1448*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 985*sin(d*x + c)^4/ (cos(d*x + c) + 1)^4 - 820*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1355*sin( d*x + c)^6/(cos(d*x + c) + 1)^6 - 520*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5)/(a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 4*a^2*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 + 5*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 5*a^2*sin(d* x + c)^6/(cos(d*x + c) + 1)^6 - 4*a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 5*(8*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a^2 + 180*log(sin(d*x + c)/ (cos(d*x + c) + 1))/a^2)/d
Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.26 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {180 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {5 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{4}} - \frac {10}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {5 \, {\left (54 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (245 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 211\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{40 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/40*(180*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 + 5*(a^2*tan(1/2*d*x + 1/2*c) ^2 - 8*a^2*tan(1/2*d*x + 1/2*c))/a^4 - 10/(a^2*(tan(1/2*d*x + 1/2*c) - 1)) - 5*(54*tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 1)/(a^2*tan(1/2 *d*x + 1/2*c)^2) + 2*(245*tan(1/2*d*x + 1/2*c)^4 + 870*tan(1/2*d*x + 1/2*c )^3 + 1240*tan(1/2*d*x + 1/2*c)^2 + 810*tan(1/2*d*x + 1/2*c) + 211)/(a^2*( tan(1/2*d*x + 1/2*c) + 1)^5))/d
Time = 32.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}+\frac {9\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {271\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{8}-\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {197\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8}+\frac {181\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {567\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{40}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{8}\right )}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \] Input:
int(1/(cos(c + d*x)^2*sin(c + d*x)^3*(a + a*sin(c + d*x))^2),x)
Output:
tan(c/2 + (d*x)/2)^2/(8*a^2*d) + (9*log(tan(c/2 + (d*x)/2)))/(2*a^2*d) - t an(c/2 + (d*x)/2)/(a^2*d) - (cot(c/2 + (d*x)/2)^2*(tan(c/2 + (d*x)/2)/2 + (567*tan(c/2 + (d*x)/2)^2)/40 + (181*tan(c/2 + (d*x)/2)^3)/5 + (197*tan(c/ 2 + (d*x)/2)^4)/8 - (41*tan(c/2 + (d*x)/2)^5)/2 - (271*tan(c/2 + (d*x)/2)^ 6)/8 - 13*tan(c/2 + (d*x)/2)^7 - 1/8))/(a^2*d*(tan(c/2 + (d*x)/2) - 1)*(ta n(c/2 + (d*x)/2) + 1)^5)
Time = 0.17 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+90 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+34 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+68 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+34 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-64 \sin \left (d x +c \right )^{5}-83 \sin \left (d x +c \right )^{4}+58 \sin \left (d x +c \right )^{3}+94 \sin \left (d x +c \right )^{2}+10 \sin \left (d x +c \right )-5}{10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(csc(d*x+c)^3*sec(d*x+c)^2/(a+a*sin(d*x+c))^2,x)
Output:
(45*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4 + 90*cos(c + d*x)*l og(tan((c + d*x)/2))*sin(c + d*x)**3 + 45*cos(c + d*x)*log(tan((c + d*x)/2 ))*sin(c + d*x)**2 + 34*cos(c + d*x)*sin(c + d*x)**4 + 68*cos(c + d*x)*sin (c + d*x)**3 + 34*cos(c + d*x)*sin(c + d*x)**2 - 64*sin(c + d*x)**5 - 83*s in(c + d*x)**4 + 58*sin(c + d*x)**3 + 94*sin(c + d*x)**2 + 10*sin(c + d*x) - 5)/(10*cos(c + d*x)*sin(c + d*x)**2*a**2*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))