Integrand size = 27, antiderivative size = 137 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}-\frac {5 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^7(c+d x)}{7 d} \] Output:
a^3*ln(sin(d*x+c))/d+3*a^3*sin(d*x+c)/d+1/2*a^3*sin(d*x+c)^2/d-5/3*a^3*sin (d*x+c)^3/d-5/4*a^3*sin(d*x+c)^4/d+1/5*a^3*sin(d*x+c)^5/d+1/2*a^3*sin(d*x+ c)^6/d+1/7*a^3*sin(d*x+c)^7/d
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (420 \log (\sin (c+d x))+1260 \sin (c+d x)+210 \sin ^2(c+d x)-700 \sin ^3(c+d x)-525 \sin ^4(c+d x)+84 \sin ^5(c+d x)+210 \sin ^6(c+d x)+60 \sin ^7(c+d x)\right )}{420 d} \] Input:
Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(420*Log[Sin[c + d*x]] + 1260*Sin[c + d*x] + 210*Sin[c + d*x]^2 - 700 *Sin[c + d*x]^3 - 525*Sin[c + d*x]^4 + 84*Sin[c + d*x]^5 + 210*Sin[c + d*x ]^6 + 60*Sin[c + d*x]^7))/(420*d)
Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 99, 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.
| \(\displaystyle \int \cos ^4(c+d x) \cot (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^3}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \csc (c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc (c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5}{a}d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\sin ^6(c+d x) a^6+3 \sin ^5(c+d x) a^6+\sin ^4(c+d x) a^6-5 \sin ^3(c+d x) a^6-5 \sin ^2(c+d x) a^6+\csc (c+d x) a^6+\sin (c+d x) a^6+3 a^6\right )d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{7} a^7 \sin ^7(c+d x)+\frac {1}{2} a^7 \sin ^6(c+d x)+\frac {1}{5} a^7 \sin ^5(c+d x)-\frac {5}{4} a^7 \sin ^4(c+d x)-\frac {5}{3} a^7 \sin ^3(c+d x)+\frac {1}{2} a^7 \sin ^2(c+d x)+3 a^7 \sin (c+d x)+a^7 \log (a \sin (c+d x))}{a^4 d}\) |
Input:
Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^7*Log[a*Sin[c + d*x]] + 3*a^7*Sin[c + d*x] + (a^7*Sin[c + d*x]^2)/2 - ( 5*a^7*Sin[c + d*x]^3)/3 - (5*a^7*Sin[c + d*x]^4)/4 + (a^7*Sin[c + d*x]^5)/ 5 + (a^7*Sin[c + d*x]^6)/2 + (a^7*Sin[c + d*x]^7)/7)/(a^4*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 9.67 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a^{3} \cos \left (d x +c \right )^{6}}{2}+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(131\) |
| default | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a^{3} \cos \left (d x +c \right )^{6}}{2}+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(131\) |
| risch | \(-i a^{3} x +\frac {9 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {2 i a^{3} c}{d}+\frac {9 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {125 a^{3} \sin \left (d x +c \right )}{64 d}-\frac {a^{3} \sin \left (7 d x +7 c \right )}{448 d}-\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {9 a^{3} \sin \left (5 d x +5 c \right )}{320 d}-\frac {a^{3} \cos \left (4 d x +4 c \right )}{16 d}+\frac {59 a^{3} \sin \left (3 d x +3 c \right )}{192 d}\) | \(171\) |
Input:
int(cos(d*x+c)^4*cot(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+ c)^2)*sin(d*x+c))-1/2*a^3*cos(d*x+c)^6+3/5*a^3*(8/3+cos(d*x+c)^4+4/3*cos(d *x+c)^2)*sin(d*x+c)+a^3*(1/4*cos(d*x+c)^4+1/2*cos(d*x+c)^2+ln(sin(d*x+c))) )
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {210 \, a^{3} \cos \left (d x + c\right )^{6} - 105 \, a^{3} \cos \left (d x + c\right )^{4} - 210 \, a^{3} \cos \left (d x + c\right )^{2} - 420 \, a^{3} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{6} - 66 \, a^{3} \cos \left (d x + c\right )^{4} - 88 \, a^{3} \cos \left (d x + c\right )^{2} - 176 \, a^{3}\right )} \sin \left (d x + c\right )}{420 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/420*(210*a^3*cos(d*x + c)^6 - 105*a^3*cos(d*x + c)^4 - 210*a^3*cos(d*x + c)^2 - 420*a^3*log(1/2*sin(d*x + c)) + 4*(15*a^3*cos(d*x + c)^6 - 66*a^3 *cos(d*x + c)^4 - 88*a^3*cos(d*x + c)^2 - 176*a^3)*sin(d*x + c))/d
\[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos ^{4}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)**4*cot(d*x+c)*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cos(c + d*x)**4*cot(c + d*x), x) + Integral(3*sin(c + d*x)* cos(c + d*x)**4*cot(c + d*x), x) + Integral(3*sin(c + d*x)**2*cos(c + d*x) **4*cot(c + d*x), x) + Integral(sin(c + d*x)**3*cos(c + d*x)**4*cot(c + d* x), x))
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/420*(60*a^3*sin(d*x + c)^7 + 210*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c )^5 - 525*a^3*sin(d*x + c)^4 - 700*a^3*sin(d*x + c)^3 + 210*a^3*sin(d*x + c)^2 + 420*a^3*log(sin(d*x + c)) + 1260*a^3*sin(d*x + c))/d
Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/420*(60*a^3*sin(d*x + c)^7 + 210*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c )^5 - 525*a^3*sin(d*x + c)^4 - 700*a^3*sin(d*x + c)^3 + 210*a^3*sin(d*x + c)^2 + 420*a^3*log(abs(sin(d*x + c))) + 1260*a^3*sin(d*x + c))/d
Time = 32.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.30 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {176\,a^3\,\sin \left (c+d\,x\right )}{105\,d}-\frac {a^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^4}{4\,d}-\frac {a^3\,{\cos \left (c+d\,x\right )}^6}{2\,d}+\frac {88\,a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{105\,d}+\frac {22\,a^3\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}-\frac {a^3\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \] Input:
int(cos(c + d*x)^4*cot(c + d*x)*(a + a*sin(c + d*x))^3,x)
Output:
(176*a^3*sin(c + d*x))/(105*d) - (a^3*log(1/cos(c/2 + (d*x)/2)^2))/d + (a^ 3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (a^3*cos(c + d*x)^2)/(2* d) + (a^3*cos(c + d*x)^4)/(4*d) - (a^3*cos(c + d*x)^6)/(2*d) + (88*a^3*cos (c + d*x)^2*sin(c + d*x))/(105*d) + (22*a^3*cos(c + d*x)^4*sin(c + d*x))/( 35*d) - (a^3*cos(c + d*x)^6*sin(c + d*x))/(7*d)
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77 \[ \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60 \sin \left (d x +c \right )^{7}+210 \sin \left (d x +c \right )^{6}+84 \sin \left (d x +c \right )^{5}-525 \sin \left (d x +c \right )^{4}-700 \sin \left (d x +c \right )^{3}+210 \sin \left (d x +c \right )^{2}+1260 \sin \left (d x +c \right )\right )}{420 d} \] Input:
int(cos(d*x+c)^4*cot(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 420*log(tan((c + d*x)/2)**2 + 1) + 420*log(tan((c + d*x)/2)) + 6 0*sin(c + d*x)**7 + 210*sin(c + d*x)**6 + 84*sin(c + d*x)**5 - 525*sin(c + d*x)**4 - 700*sin(c + d*x)**3 + 210*sin(c + d*x)**2 + 1260*sin(c + d*x))) /(420*d)