Integrand size = 21, antiderivative size = 135 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {8 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {8 \csc ^3(c+d x)}{3 a^4 d}-\frac {7 \csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^5(c+d x)}{5 a^4 d}-\frac {\csc ^6(c+d x)}{6 a^4 d}+\frac {8 \log (\sin (c+d x))}{a^4 d}-\frac {8 \log (1+\sin (c+d x))}{a^4 d} \] Output:
8*csc(d*x+c)/a^4/d-4*csc(d*x+c)^2/a^4/d+8/3*csc(d*x+c)^3/a^4/d-7/4*csc(d*x +c)^4/a^4/d+4/5*csc(d*x+c)^5/a^4/d-1/6*csc(d*x+c)^6/a^4/d+8*ln(sin(d*x+c)) /a^4/d-8*ln(1+sin(d*x+c))/a^4/d
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {480 \csc (c+d x)-240 \csc ^2(c+d x)+160 \csc ^3(c+d x)-105 \csc ^4(c+d x)+48 \csc ^5(c+d x)-10 \csc ^6(c+d x)+480 \log (\sin (c+d x))-480 \log (1+\sin (c+d x))}{60 a^4 d} \] Input:
Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]
Output:
(480*Csc[c + d*x] - 240*Csc[c + d*x]^2 + 160*Csc[c + d*x]^3 - 105*Csc[c + d*x]^4 + 48*Csc[c + d*x]^5 - 10*Csc[c + d*x]^6 + 480*Log[Sin[c + d*x]] - 4 80*Log[1 + Sin[c + d*x]])/(60*a^4*d)
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \frac {\cot ^7(c+d x)}{(a \sin (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^7 (a \sin (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {\csc ^7(c+d x) (a-a \sin (c+d x))^3}{a^7 (\sin (c+d x) a+a)}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\frac {\csc ^7(c+d x)}{a^5}-\frac {4 \csc ^6(c+d x)}{a^5}+\frac {7 \csc ^5(c+d x)}{a^5}-\frac {8 \csc ^4(c+d x)}{a^5}+\frac {8 \csc ^3(c+d x)}{a^5}-\frac {8 \csc ^2(c+d x)}{a^5}+\frac {8 \csc (c+d x)}{a^5}-\frac {8}{a^4 (\sin (c+d x) a+a)}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\csc ^6(c+d x)}{6 a^4}+\frac {4 \csc ^5(c+d x)}{5 a^4}-\frac {7 \csc ^4(c+d x)}{4 a^4}+\frac {8 \csc ^3(c+d x)}{3 a^4}-\frac {4 \csc ^2(c+d x)}{a^4}+\frac {8 \csc (c+d x)}{a^4}+\frac {8 \log (a \sin (c+d x))}{a^4}-\frac {8 \log (a \sin (c+d x)+a)}{a^4}}{d}\) |
Input:
Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]
Output:
((8*Csc[c + d*x])/a^4 - (4*Csc[c + d*x]^2)/a^4 + (8*Csc[c + d*x]^3)/(3*a^4 ) - (7*Csc[c + d*x]^4)/(4*a^4) + (4*Csc[c + d*x]^5)/(5*a^4) - Csc[c + d*x] ^6/(6*a^4) + (8*Log[a*Sin[c + d*x]])/a^4 - (8*Log[a + a*Sin[c + d*x]])/a^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 = 53.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {4}{5 \sin \left (d x +c \right )^{5}}-\frac {7}{4 \sin \left (d x +c \right )^{4}}+\frac {8}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {8}{\sin \left (d x +c \right )}+8 \ln \left (\sin \left (d x +c \right )\right )-8 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\) | \(89\) |
| default | \(\frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {4}{5 \sin \left (d x +c \right )^{5}}-\frac {7}{4 \sin \left (d x +c \right )^{4}}+\frac {8}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {8}{\sin \left (d x +c \right )}+8 \ln \left (\sin \left (d x +c \right )\right )-8 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\) | \(89\) |
| risch | \(\frac {4 i \left (-60 i {\mathrm e}^{10 i \left (d x +c \right )}+60 \,{\mathrm e}^{11 i \left (d x +c \right )}+345 i {\mathrm e}^{8 i \left (d x +c \right )}-380 \,{\mathrm e}^{9 i \left (d x +c \right )}-610 i {\mathrm e}^{6 i \left (d x +c \right )}+936 \,{\mathrm e}^{7 i \left (d x +c \right )}+345 i {\mathrm e}^{4 i \left (d x +c \right )}-936 \,{\mathrm e}^{5 i \left (d x +c \right )}-60 i {\mathrm e}^{2 i \left (d x +c \right )}+380 \,{\mathrm e}^{3 i \left (d x +c \right )}-60 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {16 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{4} d}+\frac {8 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}\) | \(192\) |
Input:
int(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d/a^4*(-1/6/sin(d*x+c)^6+4/5/sin(d*x+c)^5-7/4/sin(d*x+c)^4+8/3/sin(d*x+c )^3-4/sin(d*x+c)^2+8/sin(d*x+c)+8*ln(sin(d*x+c))-8*ln(1+sin(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.38 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {240 \, \cos \left (d x + c\right )^{4} - 585 \, \cos \left (d x + c\right )^{2} + 480 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 480 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 16 \, {\left (30 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 43\right )} \sin \left (d x + c\right ) + 355}{60 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )}} \] Input:
integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/60*(240*cos(d*x + c)^4 - 585*cos(d*x + c)^2 + 480*(cos(d*x + c)^6 - 3*co s(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c)) - 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1) - 16*(30*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 43)*sin(d*x + c) + 355)/(a^4*d *cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)
\[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\int \frac {\cot ^{7}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(cot(d*x+c)**7/(a+a*sin(d*x+c))**4,x)
Output:
Integral(cot(c + d*x)**7/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a**4
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {480 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} - \frac {480 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}} - \frac {480 \, \sin \left (d x + c\right )^{5} - 240 \, \sin \left (d x + c\right )^{4} + 160 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 48 \, \sin \left (d x + c\right ) - 10}{a^{4} \sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-1/60*(480*log(sin(d*x + c) + 1)/a^4 - 480*log(sin(d*x + c))/a^4 - (480*si n(d*x + c)^5 - 240*sin(d*x + c)^4 + 160*sin(d*x + c)^3 - 105*sin(d*x + c)^ 2 + 48*sin(d*x + c) - 10)/(a^4*sin(d*x + c)^6))/d
Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {8 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} d} + \frac {8 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4} d} + \frac {480 \, \sin \left (d x + c\right )^{5} - 240 \, \sin \left (d x + c\right )^{4} + 160 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 48 \, \sin \left (d x + c\right ) - 10}{60 \, a^{4} d \sin \left (d x + c\right )^{6}} \] Input:
integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
-8*log(abs(sin(d*x + c) + 1))/(a^4*d) + 8*log(abs(sin(d*x + c)))/(a^4*d) + 1/60*(480*sin(d*x + c)^5 - 240*sin(d*x + c)^4 + 160*sin(d*x + c)^3 - 105* sin(d*x + c)^2 + 48*sin(d*x + c) - 10)/(a^4*d*sin(d*x + c)^6)
Time = 17.73 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4\,d}-\frac {189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,a^4\,d}+\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {16\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {21\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^4\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (336\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {88\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}-\frac {1}{6}\right )}{64\,a^4\,d} \] Input:
int(cot(c + d*x)^7/(a + a*sin(c + d*x))^4,x)
Output:
(11*tan(c/2 + (d*x)/2)^3)/(24*a^4*d) - (189*tan(c/2 + (d*x)/2)^2)/(128*a^4 *d) - tan(c/2 + (d*x)/2)^4/(8*a^4*d) + tan(c/2 + (d*x)/2)^5/(40*a^4*d) - t an(c/2 + (d*x)/2)^6/(384*a^4*d) + (8*log(tan(c/2 + (d*x)/2)))/(a^4*d) - (1 6*log(tan(c/2 + (d*x)/2) + 1))/(a^4*d) + (21*tan(c/2 + (d*x)/2))/(4*a^4*d) + (cot(c/2 + (d*x)/2)^6*((8*tan(c/2 + (d*x)/2))/5 - 8*tan(c/2 + (d*x)/2)^ 2 + (88*tan(c/2 + (d*x)/2)^3)/3 - (189*tan(c/2 + (d*x)/2)^4)/2 + 336*tan(c /2 + (d*x)/2)^5 - 1/6))/(64*a^4*d)
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {-1920 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6}+960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}+325 \sin \left (d x +c \right )^{6}+960 \sin \left (d x +c \right )^{5}-480 \sin \left (d x +c \right )^{4}+320 \sin \left (d x +c \right )^{3}-210 \sin \left (d x +c \right )^{2}+96 \sin \left (d x +c \right )-20}{120 \sin \left (d x +c \right )^{6} a^{4} d} \] Input:
int(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x)
Output:
( - 1920*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6 + 960*log(tan((c + d*x) /2))*sin(c + d*x)**6 + 325*sin(c + d*x)**6 + 960*sin(c + d*x)**5 - 480*sin (c + d*x)**4 + 320*sin(c + d*x)**3 - 210*sin(c + d*x)**2 + 96*sin(c + d*x) - 20)/(120*sin(c + d*x)**6*a**4*d)