Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^6(c+d x)}{6 a d}-\frac {2 \csc ^7(c+d x)}{7 a d}+\frac {\csc ^8(c+d x)}{4 a d}+\frac {\csc ^9(c+d x)}{9 a d}-\frac {\csc ^{10}(c+d x)}{10 a d} \] Output:
1/5*csc(d*x+c)^5/a/d-1/6*csc(d*x+c)^6/a/d-2/7*csc(d*x+c)^7/a/d+1/4*csc(d*x +c)^8/a/d+1/9*csc(d*x+c)^9/a/d-1/10*csc(d*x+c)^10/a/d
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^5(c+d x) \left (252-210 \csc (c+d x)-360 \csc ^2(c+d x)+315 \csc ^3(c+d x)+140 \csc ^4(c+d x)-126 \csc ^5(c+d x)\right )}{1260 a d} \] Input:
Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(Csc[c + d*x]^5*(252 - 210*Csc[c + d*x] - 360*Csc[c + d*x]^2 + 315*Csc[c + d*x]^3 + 140*Csc[c + d*x]^4 - 126*Csc[c + d*x]^5))/(1260*a*d)
Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
\(\displaystyle \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^7}{\sin (c+d x)^{11} (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \csc ^{11}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^4 \int \frac {\csc ^{11}(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}{a^{11}}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^4 \int \left (\frac {\csc ^{11}(c+d x)}{a^6}-\frac {\csc ^{10}(c+d x)}{a^6}-\frac {2 \csc ^9(c+d x)}{a^6}+\frac {2 \csc ^8(c+d x)}{a^6}+\frac {\csc ^7(c+d x)}{a^6}-\frac {\csc ^6(c+d x)}{a^6}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^4 \left (-\frac {\csc ^{10}(c+d x)}{10 a^5}+\frac {\csc ^9(c+d x)}{9 a^5}+\frac {\csc ^8(c+d x)}{4 a^5}-\frac {2 \csc ^7(c+d x)}{7 a^5}-\frac {\csc ^6(c+d x)}{6 a^5}+\frac {\csc ^5(c+d x)}{5 a^5}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]^7*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(a^4*(Csc[c + d*x]^5/(5*a^5) - Csc[c + d*x]^6/(6*a^5) - (2*Csc[c + d*x]^7) /(7*a^5) + Csc[c + d*x]^8/(4*a^5) + Csc[c + d*x]^9/(9*a^5) - Csc[c + d*x]^ 10/(10*a^5)))/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 = 2.82 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\csc \left (d x +c \right )^{10}}{10}+\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{4}-\frac {2 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {\csc \left (d x +c \right )^{5}}{5}}{d a}\) | \(69\) |
default | \(\frac {-\frac {\csc \left (d x +c \right )^{10}}{10}+\frac {\csc \left (d x +c \right )^{9}}{9}+\frac {\csc \left (d x +c \right )^{8}}{4}-\frac {2 \csc \left (d x +c \right )^{7}}{7}-\frac {\csc \left (d x +c \right )^{6}}{6}+\frac {\csc \left (d x +c \right )^{5}}{5}}{d a}\) | \(69\) |
risch | \(\frac {32 i \left (-105 i {\mathrm e}^{14 i \left (d x +c \right )}+63 \,{\mathrm e}^{15 i \left (d x +c \right )}-210 i {\mathrm e}^{12 i \left (d x +c \right )}+45 \,{\mathrm e}^{13 i \left (d x +c \right )}-378 i {\mathrm e}^{10 i \left (d x +c \right )}+110 \,{\mathrm e}^{11 i \left (d x +c \right )}-210 i {\mathrm e}^{8 i \left (d x +c \right )}-110 \,{\mathrm e}^{9 i \left (d x +c \right )}-105 i {\mathrm e}^{6 i \left (d x +c \right )}-45 \,{\mathrm e}^{7 i \left (d x +c \right )}-63 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{315 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\) | \(150\) |
Input:
int(cot(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-1/10*csc(d*x+c)^10+1/9*csc(d*x+c)^9+1/4*csc(d*x+c)^8-2/7*csc(d*x+c )^7-1/6*csc(d*x+c)^6+1/5*csc(d*x+c)^5)
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.10 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {210 \, \cos \left (d x + c\right )^{4} - 105 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 21}{1260 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/1260*(210*cos(d*x + c)^4 - 105*cos(d*x + c)^2 - 4*(63*cos(d*x + c)^4 - 3 6*cos(d*x + c)^2 + 8)*sin(d*x + c) + 21)/(a*d*cos(d*x + c)^10 - 5*a*d*cos( d*x + c)^8 + 10*a*d*cos(d*x + c)^6 - 10*a*d*cos(d*x + c)^4 + 5*a*d*cos(d*x + c)^2 - a*d)
Timed out. \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**7*csc(d*x+c)**4/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/1260*(252*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 360*sin(d*x + c)^3 + 315 *sin(d*x + c)^2 + 140*sin(d*x + c) - 126)/(a*d*sin(d*x + c)^10)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \] Input:
integrate(cot(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/1260*(252*sin(d*x + c)^5 - 210*sin(d*x + c)^4 - 360*sin(d*x + c)^3 + 315 *sin(d*x + c)^2 + 140*sin(d*x + c) - 126)/(a*d*sin(d*x + c)^10)
Time = 33.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {252\,{\sin \left (c+d\,x\right )}^5-210\,{\sin \left (c+d\,x\right )}^4-360\,{\sin \left (c+d\,x\right )}^3+315\,{\sin \left (c+d\,x\right )}^2+140\,\sin \left (c+d\,x\right )-126}{1260\,a\,d\,{\sin \left (c+d\,x\right )}^{10}} \] Input:
int(cot(c + d*x)^7/(sin(c + d*x)^4*(a + a*sin(c + d*x))),x)
Output:
(140*sin(c + d*x) + 315*sin(c + d*x)^2 - 360*sin(c + d*x)^3 - 210*sin(c + d*x)^4 + 252*sin(c + d*x)^5 - 126)/(1260*a*d*sin(c + d*x)^10)
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {21 \sin \left (d x +c \right )^{10}+504 \sin \left (d x +c \right )^{5}-420 \sin \left (d x +c \right )^{4}-720 \sin \left (d x +c \right )^{3}+630 \sin \left (d x +c \right )^{2}+280 \sin \left (d x +c \right )-252}{2520 \sin \left (d x +c \right )^{10} a d} \] Input:
int(cot(d*x+c)^7*csc(d*x+c)^4/(a+a*sin(d*x+c)),x)
Output:
(21*sin(c + d*x)**10 + 504*sin(c + d*x)**5 - 420*sin(c + d*x)**4 - 720*sin (c + d*x)**3 + 630*sin(c + d*x)**2 + 280*sin(c + d*x) - 252)/(2520*sin(c + d*x)**10*a*d)