Integrand size = 32, antiderivative size = 78 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-2 a^3 A x-\frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d} \] Output:
-2*a^3*A*x-1/2*a^3*A*arctanh(cos(d*x+c))/d+a^3*A*cos(d*x+c)/d-2*a^3*A*cot( d*x+c)/d-1/2*a^3*A*cot(d*x+c)*csc(d*x+c)/d
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.82 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-2 a^3 A x+\frac {a^3 A \cos (c) \cos (d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a^3 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a^3 A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 A \sin (c) \sin (d x)}{d} \] Input:
Integrate[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]
Output:
-2*a^3*A*x + (a^3*A*Cos[c]*Cos[d*x])/d - (2*a^3*A*Cot[c + d*x])/d - (a^3*A *Csc[(c + d*x)/2]^2)/(8*d) - (a^3*A*Log[Cos[(c + d*x)/2]])/(2*d) + (a^3*A* Log[Sin[(c + d*x)/2]])/(2*d) + (a^3*A*Sec[(c + d*x)/2]^2)/(8*d) - (a^3*A*S in[c]*Sin[d*x])/d
Time = 0.34 (sec) , antiderivative size = 78, 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.
| \(\displaystyle \int \csc ^3(c+d x) (a \sin (c+d x)+a)^3 (A-A \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3 (A-A \sin (c+d x))}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \int \left (-a^3 A \sin (c+d x)+a^3 A \csc ^3(c+d x)+2 a^3 A \csc ^2(c+d x)-2 a^3 A\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x\) |
Input:
Int[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]),x]
Output:
-2*a^3*A*x - (a^3*A*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*A*Cos[c + d*x])/d - (2*a^3*A*Cot[c + d*x])/d - (a^3*A*Cot[c + d*x]*Csc[c + d*x])/(2*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 = 0.77 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {A \cos \left (d x +c \right ) a^{3}-2 a^{3} A \left (d x +c \right )-2 a^{3} A \cot \left (d x +c \right )+a^{3} A \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(78\) |
| default | \(\frac {A \cos \left (d x +c \right ) a^{3}-2 a^{3} A \left (d x +c \right )-2 a^{3} A \cot \left (d x +c \right )+a^{3} A \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(78\) |
| parallelrisch | \(\frac {a^{3} A \left (-16 d x +4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-8 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (d x +c \right )+8 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-17 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+23 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(126\) |
| risch | \(-2 a^{3} A x +\frac {a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a^{3} A \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-4 i {\mathrm e}^{2 i \left (d x +c \right )}+4 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(141\) |
| norman | \(\frac {\frac {a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}-\frac {a^{3} A}{8 d}+\frac {3 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}+\frac {5 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d}+\frac {27 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}-\frac {a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {3 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {2 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {2 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {3 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 d}-2 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-8 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-12 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-8 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-2 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {a^{3} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(364\) |
Input:
int(csc(d*x+c)^3*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x,method=_RETURNVERBO SE)
Output:
1/d*(A*cos(d*x+c)*a^3-2*a^3*A*(d*x+c)-2*a^3*A*cot(d*x+c)+a^3*A*(-1/2*csc(d *x+c)*cot(d*x+c)+1/2*ln(csc(d*x+c)-cot(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (74) = 148\).
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.95 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {8 \, A a^{3} d x \cos \left (d x + c\right )^{2} - 4 \, A a^{3} \cos \left (d x + c\right )^{3} - 8 \, A a^{3} d x - 8 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a^{3} \cos \left (d x + c\right ) + {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="f ricas")
Output:
-1/4*(8*A*a^3*d*x*cos(d*x + c)^2 - 4*A*a^3*cos(d*x + c)^3 - 8*A*a^3*d*x - 8*A*a^3*cos(d*x + c)*sin(d*x + c) + 2*A*a^3*cos(d*x + c) + (A*a^3*cos(d*x + c)^2 - A*a^3)*log(1/2*cos(d*x + c) + 1/2) - (A*a^3*cos(d*x + c)^2 - A*a^ 3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - d)
\[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=- A a^{3} \left (\int \left (- 2 \sin {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\right )\, dx + \int 2 \sin ^{3}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \left (- \csc ^{3}{\left (c + d x \right )}\right )\, dx\right ) \] Input:
integrate(csc(d*x+c)**3*(a+a*sin(d*x+c))**3*(A-A*sin(d*x+c)),x)
Output:
-A*a**3*(Integral(-2*sin(c + d*x)*csc(c + d*x)**3, x) + Integral(2*sin(c + d*x)**3*csc(c + d*x)**3, x) + Integral(sin(c + d*x)**4*csc(c + d*x)**3, x ) + Integral(-csc(c + d*x)**3, x))
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.15 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {8 \, {\left (d x + c\right )} A a^{3} - A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, A a^{3} \cos \left (d x + c\right ) + \frac {8 \, A a^{3}}{\tan \left (d x + c\right )}}{4 \, d} \] Input:
integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="m axima")
Output:
-1/4*(8*(d*x + c)*A*a^3 - 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)) - 4*A*a^3*cos(d*x + c) + 8*A*a ^3/tan(d*x + c))/d
Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.76 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, {\left (d x + c\right )} A a^{3} + 4 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {16 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \] Input:
integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x, algorithm="g iac")
Output:
1/8*(A*a^3*tan(1/2*d*x + 1/2*c)^2 - 16*(d*x + c)*A*a^3 + 4*A*a^3*log(abs(t an(1/2*d*x + 1/2*c))) + 8*A*a^3*tan(1/2*d*x + 1/2*c) + 16*A*a^3/(tan(1/2*d *x + 1/2*c)^2 + 1) - (6*A*a^3*tan(1/2*d*x + 1/2*c)^2 + 8*A*a^3*tan(1/2*d*x + 1/2*c) + A*a^3)/tan(1/2*d*x + 1/2*c)^2)/d
Time = 32.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.82 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {A\,a^3\,\left (\frac {\cos \left (c+d\,x\right )}{2}-4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\left (4\right )+\frac {d\,x}{2}\right )}\right )-\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\cos \left (2\,c+2\,d\,x\right )+\frac {\cos \left (3\,c+3\,d\,x\right )}{2}+2\,\sin \left (2\,c+2\,d\,x\right )+4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\left (4\right )+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )+\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2}-1\right )}{2\,d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )} \] Input:
int(((A - A*sin(c + d*x))*(a + a*sin(c + d*x))^3)/sin(c + d*x)^3,x)
Output:
(A*a^3*(cos(c + d*x)/2 - 4*atan((17^(1/2)*(4*cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2)))/(17*cos(c/2 - atan(4) + (d*x)/2))) - log(sin(c/2 + (d*x)/2)/c os(c/2 + (d*x)/2))/2 + cos(2*c + 2*d*x) + cos(3*c + 3*d*x)/2 + 2*sin(2*c + 2*d*x) + 4*atan((17^(1/2)*(4*cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2)))/(1 7*cos(c/2 - atan(4) + (d*x)/2)))*cos(2*c + 2*d*x) + (log(sin(c/2 + (d*x)/2 )/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x))/2 - 1))/(2*d*(cos(c + d*x)^2 - 1))
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.24 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {a^{4} \left (8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )-4 \cos \left (d x +c \right )+4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-16 \sin \left (d x +c \right )^{2} d x -7 \sin \left (d x +c \right )^{2}\right )}{8 \sin \left (d x +c \right )^{2} d} \] Input:
int(csc(d*x+c)^3*(a+a*sin(d*x+c))^3*(A-A*sin(d*x+c)),x)
Output:
(a**4*(8*cos(c + d*x)*sin(c + d*x)**2 - 16*cos(c + d*x)*sin(c + d*x) - 4*c os(c + d*x) + 4*log(tan((c + d*x)/2))*sin(c + d*x)**2 - 16*sin(c + d*x)**2 *d*x - 7*sin(c + d*x)**2))/(8*sin(c + d*x)**2*d)