Integrand size = 32, antiderivative size = 130 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {3 a^3 A \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d} \] Output:
3/16*a^3*A*arctanh(cos(d*x+c))/d-2/3*a^3*A*cot(d*x+c)^3/d-2/5*a^3*A*cot(d* x+c)^5/d+3/16*a^3*A*cot(d*x+c)*csc(d*x+c)/d-5/24*a^3*A*cot(d*x+c)*csc(d*x+ c)^3/d-1/6*a^3*A*cot(d*x+c)*csc(d*x+c)^5/d
Leaf count is larger than twice the leaf count of optimal. \(306\) vs. \(2(130)=260\).
Time = 0.51 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.35 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=a^3 A \left (\frac {2 \cot \left (\frac {1}{2} (c+d x)\right )}{15 d}+\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{240 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{80 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {2 \tan \left (\frac {1}{2} (c+d x)\right )}{15 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{240 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{80 d}\right ) \] Input:
Integrate[Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]
Output:
a^3*A*((2*Cot[(c + d*x)/2])/(15*d) + (3*Csc[(c + d*x)/2]^2)/(64*d) + (Cot[ (c + d*x)/2]*Csc[(c + d*x)/2]^2)/(240*d) - Csc[(c + d*x)/2]^4/(64*d) - (Co t[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(80*d) - Csc[(c + d*x)/2]^6/(384*d) + ( 3*Log[Cos[(c + d*x)/2]])/(16*d) - (3*Log[Sin[(c + d*x)/2]])/(16*d) - (3*Se c[(c + d*x)/2]^2)/(64*d) + Sec[(c + d*x)/2]^4/(64*d) + Sec[(c + d*x)/2]^6/ (384*d) - (2*Tan[(c + d*x)/2])/(15*d) - (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/ 2])/(240*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(80*d))
Time = 0.42 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 3445, 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.
| \(\displaystyle \int \csc ^7(c+d x) (a \sin (c+d x)+a)^3 (A-A \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3 (A-A \sin (c+d x))}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \int \left (a^3 A \csc ^7(c+d x)+2 a^3 A \csc ^6(c+d x)-2 a^3 A \csc ^4(c+d x)-a^3 A \csc ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 a^3 A \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}\) |
Input:
Int[Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]
Output:
(3*a^3*A*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^3*A*Cot[c + d*x]^3)/(3*d) - (2*a^3*A*Cot[c + d*x]^5)/(5*d) + (3*a^3*A*Cot[c + d*x]*Csc[c + d*x])/(16*d ) - (5*a^3*A*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*A*Cot[c + d*x]*Csc [c + d*x]^5)/(6*d)
Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[si n[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{ a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ [m] && IntegerQ[n]
Time = 1.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {-a^{3} A \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-2 a^{3} A \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+2 a^{3} A \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+a^{3} A \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(163\) |
| default | \(\frac {-a^{3} A \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-2 a^{3} A \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+2 a^{3} A \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+a^{3} A \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(163\) |
| risch | \(-\frac {a^{3} A \left (45 \,{\mathrm e}^{11 i \left (d x +c \right )}+65 \,{\mathrm e}^{9 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}+960 i {\mathrm e}^{8 i \left (d x +c \right )}-750 \,{\mathrm e}^{5 i \left (d x +c \right )}-640 i {\mathrm e}^{6 i \left (d x +c \right )}+65 \,{\mathrm e}^{3 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}-384 i {\mathrm e}^{2 i \left (d x +c \right )}+64 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {3 a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {3 a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(171\) |
| parallelrisch | \(-\frac {A \,a^{3} \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {24 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+8 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{384 d}\) | \(173\) |
Input:
int(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x,method=_RETURNVERBO SE)
Output:
1/d*(-a^3*A*(-1/2*csc(d*x+c)*cot(d*x+c)+1/2*ln(csc(d*x+c)-cot(d*x+c)))-2*a ^3*A*(-2/3-1/3*csc(d*x+c)^2)*cot(d*x+c)+2*a^3*A*(-8/15-1/5*csc(d*x+c)^4-4/ 15*csc(d*x+c)^2)*cot(d*x+c)+a^3*A*((-1/6*csc(d*x+c)^5-5/24*csc(d*x+c)^3-5/ 16*csc(d*x+c))*cot(d*x+c)+5/16*ln(csc(d*x+c)-cot(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (118) = 236\).
Time = 0.09 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.85 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {90 \, A a^{3} \cos \left (d x + c\right )^{5} - 80 \, A a^{3} \cos \left (d x + c\right )^{3} - 90 \, A a^{3} \cos \left (d x + c\right ) - 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 64 \, {\left (2 \, A a^{3} \cos \left (d x + c\right )^{5} - 5 \, A a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="f ricas")
Output:
-1/480*(90*A*a^3*cos(d*x + c)^5 - 80*A*a^3*cos(d*x + c)^3 - 90*A*a^3*cos(d *x + c) - 45*(A*a^3*cos(d*x + c)^6 - 3*A*a^3*cos(d*x + c)^4 + 3*A*a^3*cos( d*x + c)^2 - A*a^3)*log(1/2*cos(d*x + c) + 1/2) + 45*(A*a^3*cos(d*x + c)^6 - 3*A*a^3*cos(d*x + c)^4 + 3*A*a^3*cos(d*x + c)^2 - A*a^3)*log(-1/2*cos(d *x + c) + 1/2) + 64*(2*A*a^3*cos(d*x + c)^5 - 5*A*a^3*cos(d*x + c)^3)*sin( d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**7*(a+a*sin(d*x+c))**3*(A-A*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.59 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {320 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a^{3}}{\tan \left (d x + c\right )^{3}} - \frac {64 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} A a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="m axima")
Output:
1/480*(5*A*a^3*(2*(15*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 33*cos(d*x + c) )/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) - 15*log(cos( d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) - 120*A*a^3*(2*cos(d*x + c)/(cos (d*x + c)^2 - 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 320*(3 *tan(d*x + c)^2 + 1)*A*a^3/tan(d*x + c)^3 - 64*(15*tan(d*x + c)^4 + 10*tan (d*x + c)^2 + 3)*A*a^3/tan(d*x + c)^5)/d
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (118) = 236\).
Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.86 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {882 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="g iac")
Output:
1/1920*(5*A*a^3*tan(1/2*d*x + 1/2*c)^6 + 24*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^3*tan(1/2*d*x + 1/2*c)^4 + 40*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 15*A* a^3*tan(1/2*d*x + 1/2*c)^2 - 360*A*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 24 0*A*a^3*tan(1/2*d*x + 1/2*c) + (882*A*a^3*tan(1/2*d*x + 1/2*c)^6 + 240*A*a ^3*tan(1/2*d*x + 1/2*c)^5 + 15*A*a^3*tan(1/2*d*x + 1/2*c)^4 - 40*A*a^3*tan (1/2*d*x + 1/2*c)^3 - 45*A*a^3*tan(1/2*d*x + 1/2*c)^2 - 24*A*a^3*tan(1/2*d *x + 1/2*c) - 5*A*a^3)/tan(1/2*d*x + 1/2*c)^6)/d
Time = 34.65 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.62 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {A\,a^3\,\left (5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \] Input:
int(((A - A*sin(c + d*x))*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)
Output:
-(A*a^3*(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 24*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/ 2) - 45*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 40*cos(c/2 + (d*x)/2) ^3*sin(c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 2 40*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 - 240*cos(c/2 + (d*x)/2)^7*si n(c/2 + (d*x)/2)^5 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 40*cos (c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 360*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d *x)/2)^6*sin(c/2 + (d*x)/2)^6))/(1920*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d* x)/2)^6)
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {a^{4} \left (64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+45 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-96 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )-45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}\right )}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(csc(d*x+c)^7*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x)
Output:
(a**4*(64*cos(c + d*x)*sin(c + d*x)**5 + 45*cos(c + d*x)*sin(c + d*x)**4 + 32*cos(c + d*x)*sin(c + d*x)**3 - 50*cos(c + d*x)*sin(c + d*x)**2 - 96*co s(c + d*x)*sin(c + d*x) - 40*cos(c + d*x) - 45*log(tan((c + d*x)/2))*sin(c + d*x)**6))/(240*sin(c + d*x)**6*d)