Integrand size = 19, antiderivative size = 115 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d} \] Output:
-3*a*csc(d*x+c)/d-3/2*a*csc(d*x+c)^2/d+a*csc(d*x+c)^3/d+3/4*a*csc(d*x+c)^4 /d-1/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d-a*ln(sin(d*x+c))/d-a*sin(d*x+ c)/d
Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d} \] Input:
Integrate[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]
Output:
(-3*a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^ 6)/(6*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.
| \(\displaystyle \int \cot ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{\tan (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {\csc ^7(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^4}{a^7}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\csc ^7(c+d x)+\csc ^6(c+d x)-3 \csc ^5(c+d x)-3 \csc ^4(c+d x)+3 \csc ^3(c+d x)+3 \csc ^2(c+d x)-\csc (c+d x)-1\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-a \sin (c+d x)-\frac {1}{6} a \csc ^6(c+d x)-\frac {1}{5} a \csc ^5(c+d x)+\frac {3}{4} a \csc ^4(c+d x)+a \csc ^3(c+d x)-\frac {3}{2} a \csc ^2(c+d x)-3 a \csc (c+d x)-a \log (a \sin (c+d x))}{d}\) |
Input:
Int[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]
Output:
(-3*a*Csc[c + d*x] - (3*a*Csc[c + d*x]^2)/2 + a*Csc[c + d*x]^3 + (3*a*Csc[ c + d*x]^4)/4 - (a*Csc[c + d*x]^5)/5 - (a*Csc[c + d*x]^6)/6 - a*Log[a*Sin[ c + d*x]] - a*Sin[c + d*x])/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 = 1.76 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{8}}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )\right )+a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(143\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{8}}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )\right )+a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(143\) |
| risch | \(i a x +\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i a c}{d}-\frac {2 i a \left (45 i {\mathrm e}^{10 i \left (d x +c \right )}+45 \,{\mathrm e}^{11 i \left (d x +c \right )}-90 i {\mathrm e}^{8 i \left (d x +c \right )}-165 \,{\mathrm e}^{9 i \left (d x +c \right )}+170 i {\mathrm e}^{6 i \left (d x +c \right )}+318 \,{\mathrm e}^{7 i \left (d x +c \right )}-90 i {\mathrm e}^{4 i \left (d x +c \right )}-318 \,{\mathrm e}^{5 i \left (d x +c \right )}+45 i {\mathrm e}^{2 i \left (d x +c \right )}+165 \,{\mathrm e}^{3 i \left (d x +c \right )}-45 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(212\) |
Input:
int(cot(d*x+c)^7*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/5/sin(d*x+c)^5*cos(d*x+c)^8+1/5/sin(d*x+c)^3*cos(d*x+c)^8-1/sin (d*x+c)*cos(d*x+c)^8-(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2) *sin(d*x+c))+a*(-1/6*cot(d*x+c)^6+1/4*cot(d*x+c)^4-1/2*cot(d*x+c)^2-ln(sin (d*x+c))))
Time = 0.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.37 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {90 \, a \cos \left (d x + c\right )^{4} - 135 \, a \cos \left (d x + c\right )^{2} - 60 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} - 30 \, a \cos \left (d x + c\right )^{4} + 40 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) + 55 \, a}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/60*(90*a*cos(d*x + c)^4 - 135*a*cos(d*x + c)^2 - 60*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c)) - 12*(5 *a*cos(d*x + c)^6 - 30*a*cos(d*x + c)^4 + 40*a*cos(d*x + c)^2 - 16*a)*sin( d*x + c) + 55*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c) ^2 - d)
\[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \sin {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**7*(a+a*sin(d*x+c)),x)
Output:
a*(Integral(sin(c + d*x)*cot(c + d*x)**7, x) + Integral(cot(c + d*x)**7, x ))
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.79 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right ) + \frac {180 \, a \sin \left (d x + c\right )^{5} + 90 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 45 \, a \sin \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) + 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/60*(60*a*log(sin(d*x + c)) + 60*a*sin(d*x + c) + (180*a*sin(d*x + c)^5 + 90*a*sin(d*x + c)^4 - 60*a*sin(d*x + c)^3 - 45*a*sin(d*x + c)^2 + 12*a*s in(d*x + c) + 10*a)/sin(d*x + c)^6)/d
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.80 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right ) + \frac {180 \, a \sin \left (d x + c\right )^{5} + 90 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 45 \, a \sin \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) + 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/60*(60*a*log(abs(sin(d*x + c))) + 60*a*sin(d*x + c) + (180*a*sin(d*x + c)^5 + 90*a*sin(d*x + c)^4 - 60*a*sin(d*x + c)^3 - 45*a*sin(d*x + c)^2 + 1 2*a*sin(d*x + c) + 10*a)/sin(d*x + c)^6)/d
Time = 36.88 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.32 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {19\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,\left (1920\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-1920\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{1920\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {51\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128}+\frac {35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\frac {25\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{80}-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{384}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}+\frac {a}{384}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:
int(cot(c + d*x)^7*(a + a*sin(c + d*x)),x)
Output:
(3*a*tan(c/2 + (d*x)/2)^3)/(32*d) - (29*a*tan(c/2 + (d*x)/2)^2)/(128*d) - (19*a*tan(c/2 + (d*x)/2))/(16*d) + (a*tan(c/2 + (d*x)/2)^4)/(32*d) - (a*ta n(c/2 + (d*x)/2)^5)/(160*d) - (a*tan(c/2 + (d*x)/2)^6)/(384*d) - (a*(1920* log(tan(c/2 + (d*x)/2)) - 1920*log(tan(c/2 + (d*x)/2)^2 + 1)))/(1920*d) - (cot(c/2 + (d*x)/2)^6*(a/384 + (a*tan(c/2 + (d*x)/2))/160 - (11*a*tan(c/2 + (d*x)/2)^2)/384 - (7*a*tan(c/2 + (d*x)/2)^3)/80 + (25*a*tan(c/2 + (d*x)/ 2)^4)/128 + (35*a*tan(c/2 + (d*x)/2)^5)/32 + (29*a*tan(c/2 + (d*x)/2)^6)/1 28 + (51*a*tan(c/2 + (d*x)/2)^7)/16))/(d*(tan(c/2 + (d*x)/2)^2 + 1))
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.11 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (1920 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6}-1920 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}-1920 \sin \left (d x +c \right )^{7}+1435 \sin \left (d x +c \right )^{6}-5760 \sin \left (d x +c \right )^{5}-2880 \sin \left (d x +c \right )^{4}+1920 \sin \left (d x +c \right )^{3}+1440 \sin \left (d x +c \right )^{2}-384 \sin \left (d x +c \right )-320\right )}{1920 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^7*(a+a*sin(d*x+c)),x)
Output:
(a*(1920*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6 - 1920*log(tan((c + d*x)/2))*sin(c + d*x)**6 - 1920*sin(c + d*x)**7 + 1435*sin(c + d*x)**6 - 5 760*sin(c + d*x)**5 - 2880*sin(c + d*x)**4 + 1920*sin(c + d*x)**3 + 1440*s in(c + d*x)**2 - 384*sin(c + d*x) - 320))/(1920*sin(c + d*x)**6*d)