Integrand size = 25, antiderivative size = 119 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \log (\sin (c+d x))}{d} \] Output:
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+ 3/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d-1/7*a*csc(d*x+c)^7/d-a*ln(sin(d* x+c))/d
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^7*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]
Output:
(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) + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6 )/(6*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Log[Sin[c + d*x]])/d
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
| \(\displaystyle \int \cot ^7(c+d x) \csc (c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (a \sin (c+d x)+a)}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \csc ^8(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^4d(a \sin (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {\csc ^8(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^4}{a^8}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a \int \left (\frac {\csc ^8(c+d x)}{a}+\frac {\csc ^7(c+d x)}{a}-\frac {3 \csc ^6(c+d x)}{a}-\frac {3 \csc ^5(c+d x)}{a}+\frac {3 \csc ^4(c+d x)}{a}+\frac {3 \csc ^3(c+d x)}{a}-\frac {\csc ^2(c+d x)}{a}-\frac {\csc (c+d x)}{a}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (-\log (a \sin (c+d x))-\frac {1}{7} \csc ^7(c+d x)-\frac {1}{6} \csc ^6(c+d x)+\frac {3}{5} \csc ^5(c+d x)+\frac {3}{4} \csc ^4(c+d x)-\csc ^3(c+d x)-\frac {3}{2} \csc ^2(c+d x)+\csc (c+d x)\right )}{d}\) |
Input:
Int[Cot[c + d*x]^7*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]
Output:
(a*(Csc[c + d*x] - (3*Csc[c + d*x]^2)/2 - Csc[c + d*x]^3 + (3*Csc[c + d*x] ^4)/4 + (3*Csc[c + d*x]^5)/5 - Csc[c + d*x]^6/6 - Csc[c + d*x]^7/7 - Log[a *Sin[c + d*x]]))/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 = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}-\frac {3 \csc \left (d x +c \right )^{5}}{5}-\frac {3 \csc \left (d x +c \right )^{4}}{4}+\csc \left (d x +c \right )^{3}+\frac {3 \csc \left (d x +c \right )^{2}}{2}-\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
| default | \(-\frac {a \left (\frac {\csc \left (d x +c \right )^{7}}{7}+\frac {\csc \left (d x +c \right )^{6}}{6}-\frac {3 \csc \left (d x +c \right )^{5}}{5}-\frac {3 \csc \left (d x +c \right )^{4}}{4}+\csc \left (d x +c \right )^{3}+\frac {3 \csc \left (d x +c \right )^{2}}{2}-\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
| risch | \(i a x +\frac {2 i a c}{d}+\frac {2 i a \left (105 \,{\mathrm e}^{13 i \left (d x +c \right )}-210 \,{\mathrm e}^{11 i \left (d x +c \right )}-315 i {\mathrm e}^{12 i \left (d x +c \right )}+903 \,{\mathrm e}^{9 i \left (d x +c \right )}+945 i {\mathrm e}^{10 i \left (d x +c \right )}-636 \,{\mathrm e}^{7 i \left (d x +c \right )}-1820 i {\mathrm e}^{8 i \left (d x +c \right )}+903 \,{\mathrm e}^{5 i \left (d x +c \right )}+1820 i {\mathrm e}^{6 i \left (d x +c \right )}-210 \,{\mathrm e}^{3 i \left (d x +c \right )}-945 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+315 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(203\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-a/d*(1/7*csc(d*x+c)^7+1/6*csc(d*x+c)^6-3/5*csc(d*x+c)^5-3/4*csc(d*x+c)^4+ csc(d*x+c)^3+3/2*csc(d*x+c)^2-csc(d*x+c)-ln(csc(d*x+c)))
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.45 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {420 \, a \cos \left (d x + c\right )^{6} - 840 \, a \cos \left (d x + c\right )^{4} + 672 \, a \cos \left (d x + c\right )^{2} - 420 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 35 \, {\left (18 \, a \cos \left (d x + c\right )^{4} - 27 \, a \cos \left (d x + c\right )^{2} + 11 \, a\right )} \sin \left (d x + c\right ) - 192 \, a}{420 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/420*(420*a*cos(d*x + c)^6 - 840*a*cos(d*x + c)^4 + 672*a*cos(d*x + c)^2 - 420*(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))*sin(d*x + c) + 35*(18*a*cos(d*x + c)^4 - 27*a*cos(d*x + c)^2 + 11*a)*sin(d*x + c) - 192*a)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^ 4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {420 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/420*(420*a*log(sin(d*x + c)) - (420*a*sin(d*x + c)^6 - 630*a*sin(d*x + c)^5 - 420*a*sin(d*x + c)^4 + 315*a*sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)/d
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/420*(420*a*log(abs(sin(d*x + c))) - (420*a*sin(d*x + c)^6 - 630*a*sin(d *x + c)^5 - 420*a*sin(d*x + c)^4 + 315*a*sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)/d
Time = 34.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.27 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {35\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {35\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {29\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \] Input:
int((cot(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x),x)
Output:
(35*a*cot(c/2 + (d*x)/2))/(128*d) + (35*a*tan(c/2 + (d*x)/2))/(128*d) + (a *log(tan(c/2 + (d*x)/2)^2 + 1))/d - (29*a*cot(c/2 + (d*x)/2)^2)/(128*d) - (7*a*cot(c/2 + (d*x)/2)^3)/(128*d) + (a*cot(c/2 + (d*x)/2)^4)/(32*d) + (7* a*cot(c/2 + (d*x)/2)^5)/(640*d) - (a*cot(c/2 + (d*x)/2)^6)/(384*d) - (a*co t(c/2 + (d*x)/2)^7)/(896*d) - (29*a*tan(c/2 + (d*x)/2)^2)/(128*d) - (7*a*t an(c/2 + (d*x)/2)^3)/(128*d) + (a*tan(c/2 + (d*x)/2)^4)/(32*d) + (7*a*tan( c/2 + (d*x)/2)^5)/(640*d) - (a*tan(c/2 + (d*x)/2)^6)/(384*d) - (a*tan(c/2 + (d*x)/2)^7)/(896*d) - (a*log(tan(c/2 + (d*x)/2)))/d
Time = 0.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.97 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-10 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{5} \csc \left (d x +c \right )+31 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )-101 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )-60 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right ) \sin \left (d x +c \right )-60 \cot \left (d x +c \right )^{6} \csc \left (d x +c \right )+74 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right ) \sin \left (d x +c \right )+72 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )-109 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right ) \sin \left (d x +c \right )-96 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )-101 \csc \left (d x +c \right ) \sin \left (d x +c \right )+192 \csc \left (d x +c \right )+420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{420 d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)*(a+a*sin(d*x+c)),x)
Output:
(a*( - 10*cos(c + d*x)*cot(c + d*x)**5*csc(c + d*x) + 31*cos(c + d*x)*cot( c + d*x)**3*csc(c + d*x) - 101*cos(c + d*x)*cot(c + d*x)*csc(c + d*x) - 60 *cot(c + d*x)**6*csc(c + d*x)*sin(c + d*x) - 60*cot(c + d*x)**6*csc(c + d* x) + 74*cot(c + d*x)**4*csc(c + d*x)*sin(c + d*x) + 72*cot(c + d*x)**4*csc (c + d*x) - 109*cot(c + d*x)**2*csc(c + d*x)*sin(c + d*x) - 96*cot(c + d*x )**2*csc(c + d*x) - 101*csc(c + d*x)*sin(c + d*x) + 192*csc(c + d*x) + 420 *log(tan((c + d*x)/2)**2 + 1) - 420*log(tan((c + d*x)/2))))/(420*d)