Integrand size = 29, antiderivative size = 100 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d} \] Output:
7/8*a^3*arctanh(cos(d*x+c))/d-4/3*a^3*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c)^5/ d-1/8*a^3*cot(d*x+c)*csc(d*x+c)/d-3/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(100)=200\).
Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.67 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (\frac {17 \cot \left (\frac {1}{2} (c+d x)\right )}{30 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {59 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{480 d}-\frac {3 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{160 d}+\frac {7 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {7 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {17 \tan \left (\frac {1}{2} (c+d x)\right )}{30 d}+\frac {59 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{480 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{160 d}\right ) \] Input:
Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
a^3*((17*Cot[(c + d*x)/2])/(30*d) - Csc[(c + d*x)/2]^2/(32*d) - (59*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(480*d) - (3*Csc[(c + d*x)/2]^4)/(64*d) - ( Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(160*d) + (7*Log[Cos[(c + d*x)/2]])/( 8*d) - (7*Log[Sin[(c + d*x)/2]])/(8*d) + Sec[(c + d*x)/2]^2/(32*d) + (3*Se c[(c + d*x)/2]^4)/(64*d) - (17*Tan[(c + d*x)/2])/(30*d) + (59*Sec[(c + d*x )/2]^2*Tan[(c + d*x)/2])/(480*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/( 160*d))
Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {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.
| \(\displaystyle \int \cot ^2(c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2 (a \sin (c+d x)+a)^3}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^2(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)+a^3 \cot ^2(c+d x) \csc (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{8 d}\) |
Input:
Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(7*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (a^3* Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot [c + d*x]*Csc[c + d*x]^3)/(4*d)
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]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(\frac {a^{3} \left (360 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}-960 i {\mathrm e}^{6 i \left (d x +c \right )}-390 \,{\mathrm e}^{7 i \left (d x +c \right )}+400 i {\mathrm e}^{4 i \left (d x +c \right )}-320 i {\mathrm e}^{2 i \left (d x +c \right )}+390 \,{\mathrm e}^{3 i \left (d x +c \right )}+136 i-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(158\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \cos \left (d x +c \right )^{3}}{15 \sin \left (d x +c \right )^{3}}\right )}{d}\) | \(185\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \cos \left (d x +c \right )^{3}}{15 \sin \left (d x +c \right )^{3}}\right )}{d}\) | \(185\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/60*a^3*(360*I*exp(8*I*(d*x+c))+15*exp(9*I*(d*x+c))-960*I*exp(6*I*(d*x+c) )-390*exp(7*I*(d*x+c))+400*I*exp(4*I*(d*x+c))-320*I*exp(2*I*(d*x+c))+390*e xp(3*I*(d*x+c))+136*I-15*exp(I*(d*x+c)))/d/(exp(2*I*(d*x+c))-1)^5-7/8*a^3/ d*ln(exp(I*(d*x+c))-1)+7/8*a^3/d*ln(exp(I*(d*x+c))+1)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (90) = 180\).
Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.90 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {272 \, a^{3} \cos \left (d x + c\right )^{5} - 320 \, a^{3} \cos \left (d x + c\right )^{3} + 105 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - 7 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/240*(272*a^3*cos(d*x + c)^5 - 320*a^3*cos(d*x + c)^3 + 105*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 105*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d *x + c) + 1/2)*sin(d*x + c) + 30*(a^3*cos(d*x + c)^3 - 7*a^3*cos(d*x + c)) *sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.55 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {45 \, a^{3} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 60 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a^{3}}{\tan \left (d x + c\right )^{3}} + \frac {16 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{240 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/240*(45*a^3*(2*(cos(d*x + c)^3 + cos(d*x + c))/(cos(d*x + c)^4 - 2*cos( d*x + c)^2 + 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 60*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)) + 240*a^3/tan(d*x + c)^3 + 16*(5*tan(d*x + c)^2 + 3)*a^3/tan(d *x + c)^5)/d
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (90) = 180\).
Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.96 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 130 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 420 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1918 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 130 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 + 45*a^3*tan(1/2*d*x + 1/2*c)^4 + 130* a^3*tan(1/2*d*x + 1/2*c)^3 + 120*a^3*tan(1/2*d*x + 1/2*c)^2 - 840*a^3*log( abs(tan(1/2*d*x + 1/2*c))) - 420*a^3*tan(1/2*d*x + 1/2*c) + (1918*a^3*tan( 1/2*d*x + 1/2*c)^5 + 420*a^3*tan(1/2*d*x + 1/2*c)^4 - 120*a^3*tan(1/2*d*x + 1/2*c)^3 - 130*a^3*tan(1/2*d*x + 1/2*c)^2 - 45*a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 18.77 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.91 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3\,\left (6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}{960\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \] Input:
int((cot(c + d*x)^2*(a + a*sin(c + d*x))^3)/sin(c + d*x)^4,x)
Output:
-(a^3*(6*cos(c/2 + (d*x)/2)^10 - 6*sin(c/2 + (d*x)/2)^10 - 45*cos(c/2 + (d *x)/2)*sin(c/2 + (d*x)/2)^9 + 45*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2) - 130*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^8 - 120*cos(c/2 + (d*x)/2)^3* sin(c/2 + (d*x)/2)^7 + 420*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^6 - 420 *cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^4 + 120*cos(c/2 + (d*x)/2)^7*sin( c/2 + (d*x)/2)^3 + 130*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2 + 840*log (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d* x)/2)^5))/(960*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^5)
Time = 0.16 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (136 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-112 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-90 \cos \left (d x +c \right ) \sin \left (d x +c \right )-24 \cos \left (d x +c \right )-105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5}\right )}{120 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(136*cos(c + d*x)*sin(c + d*x)**4 - 15*cos(c + d*x)*sin(c + d*x)**3 - 112*cos(c + d*x)*sin(c + d*x)**2 - 90*cos(c + d*x)*sin(c + d*x) - 24*cos (c + d*x) - 105*log(tan((c + d*x)/2))*sin(c + d*x)**5))/(120*sin(c + d*x)* *5*d)