Integrand size = 25, antiderivative size = 89 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 b x}{8}-\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d} \] Output:
3/8*b*x-a*arctanh(cos(d*x+c))/d+a*cos(d*x+c)/d+1/3*a*cos(d*x+c)^3/d+3/8*b* cos(d*x+c)*sin(d*x+c)/d+1/4*b*cos(d*x+c)^3*sin(d*x+c)/d
Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.22 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {3 b (c+d x)}{8 d}+\frac {5 a \cos (c+d x)}{4 d}+\frac {a \cos (3 (c+d x))}{12 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {b \sin (2 (c+d x))}{4 d}+\frac {b \sin (4 (c+d x))}{32 d} \] Input:
Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]
Output:
(3*b*(c + d*x))/(8*d) + (5*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(1 2*d) - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (b*Sin[ 2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)
Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3317, 3042, 25, 3072, 254, 2009, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^3(c+d x) \cot (c+d x)dx+b \int \cos ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int -\sin \left (c+d x+\frac {\pi }{2}\right )^3 \tan \left (c+d x+\frac {\pi }{2}\right )dx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \int \frac {\cos ^4(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \int \left (-\cos ^2(c+d x)+\frac {1}{1-\cos ^2(c+d x)}-1\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle b \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {a \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]
Output:
-((a*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x] - Cos[c + d*x]^3/3))/d) + b*((C os[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/ (2*d)))/4)
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 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]
Time = 1.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {a \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(76\) |
| default | \(\frac {a \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(76\) |
| risch | \(\frac {3 b x}{8}+\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {b \sin \left (4 d x +4 c \right )}{32 d}+\frac {a \cos \left (3 d x +3 c \right )}{12 d}+\frac {b \sin \left (2 d x +2 c \right )}{4 d}\) | \(116\) |
Input:
int(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(1/3*cos(d*x+c)^3+cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c)))+b*(1/4*(cos (d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {8 \, a \cos \left (d x + c\right )^{3} + 9 \, b d x + 24 \, a \cos \left (d x + c\right ) - 12 \, a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 12 \, a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (2 \, b \cos \left (d x + c\right )^{3} + 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="fricas")
Output:
1/24*(8*a*cos(d*x + c)^3 + 9*b*d*x + 24*a*cos(d*x + c) - 12*a*log(1/2*cos( d*x + c) + 1/2) + 12*a*log(-1/2*cos(d*x + c) + 1/2) + 3*(2*b*cos(d*x + c)^ 3 + 3*b*cos(d*x + c))*sin(d*x + c))/d
\[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**3*cot(d*x+c)*(a+b*sin(d*x+c)),x)
Output:
Integral((a + b*sin(c + d*x))*cos(c + d*x)**3*cot(c + d*x), x)
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {16 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{96 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="maxima")
Output:
1/96*(16*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3* log(cos(d*x + c) - 1))*a + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d *x + 2*c))*b)/d
Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.63 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {9 \, {\left (d x + c\right )} b + 24 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 96 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 32 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/24*(9*(d*x + c)*b + 24*a*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(15*b*tan(1/ 2*d*x + 1/2*c)^7 - 48*a*tan(1/2*d*x + 1/2*c)^6 - 9*b*tan(1/2*d*x + 1/2*c)^ 5 - 96*a*tan(1/2*d*x + 1/2*c)^4 + 9*b*tan(1/2*d*x + 1/2*c)^3 - 80*a*tan(1/ 2*d*x + 1/2*c)^2 - 15*b*tan(1/2*d*x + 1/2*c) - 32*a)/(tan(1/2*d*x + 1/2*c) ^2 + 1)^4)/d
Time = 34.37 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.72 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {3\,b\,\mathrm {atan}\left (\frac {9\,b^2}{16\,\left (\frac {3\,a\,b}{2}-\frac {9\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {3\,a\,b}{2}-\frac {9\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {-\frac {5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {5\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {8\,a}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \] Input:
int(cos(c + d*x)^3*cot(c + d*x)*(a + b*sin(c + d*x)),x)
Output:
(3*b*atan((9*b^2)/(16*((3*a*b)/2 - (9*b^2*tan(c/2 + (d*x)/2))/16)) + (3*a* b*tan(c/2 + (d*x)/2))/(2*((3*a*b)/2 - (9*b^2*tan(c/2 + (d*x)/2))/16))))/(4 *d) + ((8*a)/3 + (5*b*tan(c/2 + (d*x)/2))/4 + (20*a*tan(c/2 + (d*x)/2)^2)/ 3 + 8*a*tan(c/2 + (d*x)/2)^4 + 4*a*tan(c/2 + (d*x)/2)^6 - (3*b*tan(c/2 + ( d*x)/2)^3)/4 + (3*b*tan(c/2 + (d*x)/2)^5)/4 - (5*b*tan(c/2 + (d*x)/2)^7)/4 )/(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/ 2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) + (a*log(tan(c/2 + (d*x)/2)))/d
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +32 \cos \left (d x +c \right ) a +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -32 a +9 b c +9 b d x}{24 d} \] Input:
int(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c)),x)
Output:
( - 6*cos(c + d*x)*sin(c + d*x)**3*b - 8*cos(c + d*x)*sin(c + d*x)**2*a + 15*cos(c + d*x)*sin(c + d*x)*b + 32*cos(c + d*x)*a + 24*log(tan((c + d*x)/ 2))*a - 32*a + 9*b*c + 9*b*d*x)/(24*d)