Integrand size = 25, antiderivative size = 88 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=b x-\frac {3 a \text {arctanh}(\cos (c+d x))}{8 d}+\frac {b \cot (c+d x)}{d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d} \] Output:
b*x-3/8*a*arctanh(cos(d*x+c))/d+b*cot(d*x+c)/d-1/3*b*cot(d*x+c)^3/d+3/8*a* cot(d*x+c)*csc(d*x+c)/d-1/4*a*cot(d*x+c)^3*csc(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]
Output:
(5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) - (3*a* Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (5*a*Se c[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)
Time = 0.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3317, 3042, 3091, 3042, 3091, 3042, 3954, 3042, 3954, 24, 4257}
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.
| \(\displaystyle \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cot ^4(c+d x) \csc (c+d x)dx+b \int \cot ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx+b \int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \left (-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \left (-\int \cot ^2(c+d x)dx-\frac {\cot ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \left (-\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \left (\int 1dx-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \left (-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}+x\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )+b \left (-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}+x\right )\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]
Output:
b*(x + Cot[c + d*x]/d - Cot[c + d*x]^3/(3*d)) + a*(-1/4*(Cot[c + d*x]^3*Cs c[c + d*x])/d - (3*(ArcTanh[Cos[c + d*x]]/(2*d) - (Cot[c + d*x]*Csc[c + d* x])/(2*d)))/4)
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(104\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(104\) |
| risch | \(b x -\frac {-48 i b \,{\mathrm e}^{6 i \left (d x +c \right )}+15 a \,{\mathrm e}^{7 i \left (d x +c \right )}+96 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 a \,{\mathrm e}^{5 i \left (d x +c \right )}-80 i b \,{\mathrm e}^{2 i \left (d x +c \right )}+9 a \,{\mathrm e}^{3 i \left (d x +c \right )}+32 i b +15 a \,{\mathrm e}^{i \left (d x +c \right )}}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(151\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/4/sin(d*x+c)^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c)^5+1/8*c os(d*x+c)^3+3/8*cos(d*x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c)))+b*(-1/3*cot(d*x+ c)^3+cot(d*x+c)+d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (80) = 160\).
Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.05 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {48 \, b d x \cos \left (d x + c\right )^{4} - 96 \, b d x \cos \left (d x + c\right )^{2} - 30 \, a \cos \left (d x + c\right )^{3} + 48 \, b d x + 18 \, a \cos \left (d x + c\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (4 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="fricas")
Output:
1/48*(48*b*d*x*cos(d*x + c)^4 - 96*b*d*x*cos(d*x + c)^2 - 30*a*cos(d*x + c )^3 + 48*b*d*x + 18*a*cos(d*x + c) - 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c )^2 + a)*log(1/2*cos(d*x + c) + 1/2) + 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(-1/2*cos(d*x + c) + 1/2) - 16*(4*b*cos(d*x + c)^3 - 3*b*cos (d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
\[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \cot ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)*(a+b*sin(d*x+c)),x)
Output:
Integral((a + b*sin(c + d*x))*cot(c + d*x)**4*csc(c + d*x), x)
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} b - 3 \, a {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="maxima")
Output:
1/48*(16*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*b - 3*a*(2* (5*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1 ) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d
Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.74 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 192 \, {\left (d x + c\right )} b + 72 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 + 8*b*tan(1/2*d*x + 1/2*c)^3 - 24*a*tan( 1/2*d*x + 1/2*c)^2 + 192*(d*x + c)*b + 72*a*log(abs(tan(1/2*d*x + 1/2*c))) - 120*b*tan(1/2*d*x + 1/2*c) - (150*a*tan(1/2*d*x + 1/2*c)^4 - 120*b*tan( 1/2*d*x + 1/2*c)^3 - 24*a*tan(1/2*d*x + 1/2*c)^2 + 8*b*tan(1/2*d*x + 1/2*c ) + 3*a)/tan(1/2*d*x + 1/2*c)^4)/d
Time = 32.70 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.51 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {3\,a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}+\frac {5\,b\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {5\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {2\,b\,\mathrm {atan}\left (\frac {8\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {b\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d} \] Input:
int((cot(c + d*x)^4*(a + b*sin(c + d*x)))/sin(c + d*x),x)
Output:
(3*a*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(8*d) + (5*b*cot(c/2 + (d *x)/2))/(8*d) - (5*b*tan(c/2 + (d*x)/2))/(8*d) + (2*b*atan((8*b*cos(c/2 + (d*x)/2) + 3*a*sin(c/2 + (d*x)/2))/(3*a*cos(c/2 + (d*x)/2) - 8*b*sin(c/2 + (d*x)/2))))/d + (a*cot(c/2 + (d*x)/2)^2)/(8*d) - (a*cot(c/2 + (d*x)/2)^4) /(64*d) - (b*cot(c/2 + (d*x)/2)^3)/(24*d) - (a*tan(c/2 + (d*x)/2)^2)/(8*d) + (a*tan(c/2 + (d*x)/2)^4)/(64*d) + (b*tan(c/2 + (d*x)/2)^3)/(24*d)
Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx=\frac {32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a -8 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -6 \cos \left (d x +c \right ) a +9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a +24 \sin \left (d x +c \right )^{4} b d x}{24 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c)),x)
Output:
(32*cos(c + d*x)*sin(c + d*x)**3*b + 15*cos(c + d*x)*sin(c + d*x)**2*a - 8 *cos(c + d*x)*sin(c + d*x)*b - 6*cos(c + d*x)*a + 9*log(tan((c + d*x)/2))* sin(c + d*x)**4*a + 24*sin(c + d*x)**4*b*d*x)/(24*sin(c + d*x)**4*d)