Integrand size = 21, antiderivative size = 148 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {4 a^4 \csc (c+d x)}{d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {4 a^4 \csc ^3(c+d x)}{3 d}-\frac {a^4 \csc ^4(c+d x)}{4 d}-\frac {10 a^4 \log (\sin (c+d x))}{d}-\frac {4 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{4 d} \] Output:
4*a^4*csc(d*x+c)/d-2*a^4*csc(d*x+c)^2/d-4/3*a^4*csc(d*x+c)^3/d-1/4*a^4*csc (d*x+c)^4/d-10*a^4*ln(sin(d*x+c))/d-4*a^4*sin(d*x+c)/d+2*a^4*sin(d*x+c)^2/ d+4/3*a^4*sin(d*x+c)^3/d+1/4*a^4*sin(d*x+c)^4/d
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \left (48 \csc (c+d x)-24 \csc ^2(c+d x)-16 \csc ^3(c+d x)-3 \csc ^4(c+d x)-120 \log (\sin (c+d x))-48 \sin (c+d x)+24 \sin ^2(c+d x)+16 \sin ^3(c+d x)+3 \sin ^4(c+d x)\right )}{12 d} \] Input:
Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]
Output:
(a^4*(48*Csc[c + d*x] - 24*Csc[c + d*x]^2 - 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 120*Log[Sin[c + d*x]] - 48*Sin[c + d*x] + 24*Sin[c + d*x]^2 + 16* Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/(12*d)
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 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.
| \(\displaystyle \int \cot ^5(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {\csc ^5(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^6}{a^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (a^3 \csc ^5(c+d x)+4 a^3 \csc ^4(c+d x)+4 a^3 \csc ^3(c+d x)-4 a^3 \csc ^2(c+d x)-10 a^3 \csc (c+d x)-4 a^3+a^3 \sin ^3(c+d x)+4 a^3 \sin ^2(c+d x)+4 a^3 \sin (c+d x)\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} a^4 \sin ^4(c+d x)+\frac {4}{3} a^4 \sin ^3(c+d x)+2 a^4 \sin ^2(c+d x)-4 a^4 \sin (c+d x)-\frac {1}{4} a^4 \csc ^4(c+d x)-\frac {4}{3} a^4 \csc ^3(c+d x)-2 a^4 \csc ^2(c+d x)+4 a^4 \csc (c+d x)-10 a^4 \log (a \sin (c+d x))}{d}\) |
Input:
Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]
Output:
(4*a^4*Csc[c + d*x] - 2*a^4*Csc[c + d*x]^2 - (4*a^4*Csc[c + d*x]^3)/3 - (a ^4*Csc[c + d*x]^4)/4 - 10*a^4*Log[a*Sin[c + d*x]] - 4*a^4*Sin[c + d*x] + 2 *a^4*Sin[c + d*x]^2 + (4*a^4*Sin[c + d*x]^3)/3 + (a^4*Sin[c + d*x]^4)/4)/d
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Time = 7.41 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.64
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(243\) |
| default | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+6 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+a^{4} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(243\) |
| risch | \(10 i a^{4} x +\frac {a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{6 d}-\frac {9 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {3 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {9 a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {i a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{6 d}+\frac {a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {20 i a^{4} c}{d}+\frac {4 i a^{4} \left (-6 i {\mathrm e}^{6 i \left (d x +c \right )}+6 \,{\mathrm e}^{7 i \left (d x +c \right )}+15 i {\mathrm e}^{4 i \left (d x +c \right )}-10 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {10 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(282\) |
Input:
int(cot(d*x+c)^5*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(a^4*(1/4*cos(d*x+c)^4+1/2*cos(d*x+c)^2+ln(sin(d*x+c)))+4*a^4*(-1/sin( d*x+c)*cos(d*x+c)^6-(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+6*a^4* (-1/2/sin(d*x+c)^2*cos(d*x+c)^6-1/2*cos(d*x+c)^4-cos(d*x+c)^2-2*ln(sin(d*x +c)))+4*a^4*(-1/3/sin(d*x+c)^3*cos(d*x+c)^6+1/sin(d*x+c)*cos(d*x+c)^6+(8/3 +cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+a^4*(-1/4*cot(d*x+c)^4+1/2*cot (d*x+c)^2+ln(sin(d*x+c))))
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {24 \, a^{4} \cos \left (d x + c\right )^{8} - 128 \, a^{4} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 288 \, a^{4} \cos \left (d x + c\right )^{6} + 615 \, a^{4} \cos \left (d x + c\right )^{4} - 270 \, a^{4} \cos \left (d x + c\right )^{2} - 105 \, a^{4} - 960 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^5*(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/96*(24*a^4*cos(d*x + c)^8 - 128*a^4*cos(d*x + c)^6*sin(d*x + c) - 288*a^ 4*cos(d*x + c)^6 + 615*a^4*cos(d*x + c)^4 - 270*a^4*cos(d*x + c)^2 - 105*a ^4 - 960*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(1/2*sin(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
\[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**5*(a+a*sin(d*x+c))**4,x)
Output:
a**4*(Integral(4*sin(c + d*x)*cot(c + d*x)**5, x) + Integral(6*sin(c + d*x )**2*cot(c + d*x)**5, x) + Integral(4*sin(c + d*x)**3*cot(c + d*x)**5, x) + Integral(sin(c + d*x)**4*cot(c + d*x)**5, x) + Integral(cot(c + d*x)**5, x))
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac {48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 24*a^4*sin(d*x + c)^2 - 120*a^4*log(sin(d*x + c)) - 48*a^4*sin(d*x + c) + (48*a^4*sin(d*x + c)^ 3 - 24*a^4*sin(d*x + c)^2 - 16*a^4*sin(d*x + c) - 3*a^4)/sin(d*x + c)^4)/d
Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac {48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 24*a^4*sin(d*x + c)^2 - 120*a^4*log(abs(sin(d*x + c))) - 48*a^4*sin(d*x + c) + (48*a^4*sin(d*x + c)^3 - 24*a^4*sin(d*x + c)^2 - 16*a^4*sin(d*x + c) - 3*a^4)/sin(d*x + c) ^4)/d
Time = 33.93 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.49 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {10\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-119\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+120\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {1135\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}+80\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-73\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+48\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {75\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {40\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a^4}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {9\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}+\frac {10\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cot(c + d*x)^5*(a + a*sin(c + d*x))^4,x)
Output:
(3*a^4*tan(c/2 + (d*x)/2))/(2*d) - (a^4*tan(c/2 + (d*x)/2)^3)/(6*d) - (a^4 *tan(c/2 + (d*x)/2)^4)/(64*d) - (10*a^4*log(tan(c/2 + (d*x)/2)))/d - (10*a ^4*tan(c/2 + (d*x)/2)^2 - (40*a^4*tan(c/2 + (d*x)/2)^3)/3 + (75*a^4*tan(c/ 2 + (d*x)/2)^4)/2 + 48*a^4*tan(c/2 + (d*x)/2)^5 - 73*a^4*tan(c/2 + (d*x)/2 )^6 + 80*a^4*tan(c/2 + (d*x)/2)^7 - (1135*a^4*tan(c/2 + (d*x)/2)^8)/4 + 12 0*a^4*tan(c/2 + (d*x)/2)^9 - 119*a^4*tan(c/2 + (d*x)/2)^10 + 104*a^4*tan(c /2 + (d*x)/2)^11 + a^4/4 + (8*a^4*tan(c/2 + (d*x)/2))/3)/(d*(16*tan(c/2 + (d*x)/2)^4 + 64*tan(c/2 + (d*x)/2)^6 + 96*tan(c/2 + (d*x)/2)^8 + 64*tan(c/ 2 + (d*x)/2)^10 + 16*tan(c/2 + (d*x)/2)^12)) - (9*a^4*tan(c/2 + (d*x)/2)^2 )/(16*d) + (10*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d
Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.95 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^{4} \left (480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4}-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+12 \sin \left (d x +c \right )^{8}+64 \sin \left (d x +c \right )^{7}+96 \sin \left (d x +c \right )^{6}-192 \sin \left (d x +c \right )^{5}+165 \sin \left (d x +c \right )^{4}+192 \sin \left (d x +c \right )^{3}-96 \sin \left (d x +c \right )^{2}-64 \sin \left (d x +c \right )-12\right )}{48 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+a*sin(d*x+c))^4,x)
Output:
(a**4*(480*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4 - 480*log(tan((c + d*x)/2))*sin(c + d*x)**4 + 12*sin(c + d*x)**8 + 64*sin(c + d*x)**7 + 96*s in(c + d*x)**6 - 192*sin(c + d*x)**5 + 165*sin(c + d*x)**4 + 192*sin(c + d *x)**3 - 96*sin(c + d*x)**2 - 64*sin(c + d*x) - 12))/(48*sin(c + d*x)**4*d )