Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 b \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {b \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \] Output:
-1/5*b*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d+2/7*b*csc(d*x+c)^7/d+1/4*a*csc( d*x+c)^8/d-1/9*b*csc(d*x+c)^9/d-1/10*a*csc(d*x+c)^10/d
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {2 b \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {b \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]
Output:
-1/5*(b*Csc[c + d*x]^5)/d - (a*Csc[c + d*x]^6)/(6*d) + (2*b*Csc[c + d*x]^7 )/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (b*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)
Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3316, 27, 522, 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.
\(\displaystyle \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^5 (a+b \sin (c+d x))}{\sin (c+d x)^{11}}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle \frac {\int \csc ^{11}(c+d x) (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2d(b \sin (c+d x))}{b^5 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^6 \int \frac {\csc ^{11}(c+d x) (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^{11}}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {b^6 \int \left (\frac {a \csc ^{11}(c+d x)}{b^7}+\frac {\csc ^{10}(c+d x)}{b^6}-\frac {2 a \csc ^9(c+d x)}{b^7}-\frac {2 \csc ^8(c+d x)}{b^6}+\frac {a \csc ^7(c+d x)}{b^7}+\frac {\csc ^6(c+d x)}{b^6}\right )d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^6 \left (-\frac {a \csc ^{10}(c+d x)}{10 b^6}+\frac {a \csc ^8(c+d x)}{4 b^6}-\frac {a \csc ^6(c+d x)}{6 b^6}-\frac {\csc ^9(c+d x)}{9 b^5}+\frac {2 \csc ^7(c+d x)}{7 b^5}-\frac {\csc ^5(c+d x)}{5 b^5}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]
Output:
(b^6*(-1/5*Csc[c + d*x]^5/b^5 - (a*Csc[c + d*x]^6)/(6*b^6) + (2*Csc[c + d* x]^7)/(7*b^5) + (a*Csc[c + d*x]^8)/(4*b^6) - Csc[c + d*x]^9/(9*b^5) - (a*C sc[c + d*x]^10)/(10*b^6)))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
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*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 0.61 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {\frac {a \csc \left (d x +c \right )^{10}}{10}+\frac {b \csc \left (d x +c \right )^{9}}{9}-\frac {a \csc \left (d x +c \right )^{8}}{4}-\frac {2 b \csc \left (d x +c \right )^{7}}{7}+\frac {a \csc \left (d x +c \right )^{6}}{6}+\frac {b \csc \left (d x +c \right )^{5}}{5}}{d}\) | \(73\) |
default | \(-\frac {\frac {a \csc \left (d x +c \right )^{10}}{10}+\frac {b \csc \left (d x +c \right )^{9}}{9}-\frac {a \csc \left (d x +c \right )^{8}}{4}-\frac {2 b \csc \left (d x +c \right )^{7}}{7}+\frac {a \csc \left (d x +c \right )^{6}}{6}+\frac {b \csc \left (d x +c \right )^{5}}{5}}{d}\) | \(73\) |
risch | \(-\frac {32 i \left (105 i a \,{\mathrm e}^{14 i \left (d x +c \right )}+63 b \,{\mathrm e}^{15 i \left (d x +c \right )}+210 i a \,{\mathrm e}^{12 i \left (d x +c \right )}+45 b \,{\mathrm e}^{13 i \left (d x +c \right )}+378 i a \,{\mathrm e}^{10 i \left (d x +c \right )}+110 b \,{\mathrm e}^{11 i \left (d x +c \right )}+210 i a \,{\mathrm e}^{8 i \left (d x +c \right )}-110 b \,{\mathrm e}^{9 i \left (d x +c \right )}+105 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-45 b \,{\mathrm e}^{7 i \left (d x +c \right )}-63 b \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}\) | \(158\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^6*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d*(1/10*a*csc(d*x+c)^10+1/9*b*csc(d*x+c)^9-1/4*a*csc(d*x+c)^8-2/7*b*csc (d*x+c)^7+1/6*a*csc(d*x+c)^6+1/5*b*csc(d*x+c)^5)
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=\frac {210 \, a \cos \left (d x + c\right )^{4} - 105 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, b \cos \left (d x + c\right )^{4} - 36 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 21 \, a}{1260 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^6*(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/1260*(210*a*cos(d*x + c)^4 - 105*a*cos(d*x + c)^2 + 4*(63*b*cos(d*x + c) ^4 - 36*b*cos(d*x + c)^2 + 8*b)*sin(d*x + c) + 21*a)/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d *x + c)^2 - d)
Timed out. \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**6*(a+b*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, b \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, b \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^6*(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/1260*(252*b*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*b*sin(d*x + c)^ 3 - 315*a*sin(d*x + c)^2 + 140*b*sin(d*x + c) + 126*a)/(d*sin(d*x + c)^10)
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252 \, b \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, b \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, b \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^6*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
-1/1260*(252*b*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*b*sin(d*x + c)^ 3 - 315*a*sin(d*x + c)^2 + 140*b*sin(d*x + c) + 126*a)/(d*sin(d*x + c)^10)
Time = 21.00 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {252\,b\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-360\,b\,{\sin \left (c+d\,x\right )}^3-315\,a\,{\sin \left (c+d\,x\right )}^2+140\,b\,\sin \left (c+d\,x\right )+126\,a}{1260\,d\,{\sin \left (c+d\,x\right )}^{10}} \] Input:
int((cot(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^6,x)
Output:
-(126*a + 140*b*sin(c + d*x) - 315*a*sin(c + d*x)^2 + 210*a*sin(c + d*x)^4 - 360*b*sin(c + d*x)^3 + 252*b*sin(c + d*x)^5)/(1260*d*sin(c + d*x)^10)
Time = 0.18 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.34 \[ \int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-56 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{5} b +61 \cos \left (d x +c \right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{5} b -504 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{6} b -504 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{5} a +259 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{6} b +252 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{5} a +61 \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{6} b -84 \csc \left (d x +c \right )^{6} \sin \left (d x +c \right )^{5} a -189 b}{5040 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^6*(a+b*sin(d*x+c)),x)
Output:
( - 56*cos(c + d*x)*cot(c + d*x)**3*csc(c + d*x)**6*sin(c + d*x)**5*b + 61 *cos(c + d*x)*cot(c + d*x)*csc(c + d*x)**6*sin(c + d*x)**5*b - 504*cot(c + d*x)**4*csc(c + d*x)**6*sin(c + d*x)**6*b - 504*cot(c + d*x)**4*csc(c + d *x)**6*sin(c + d*x)**5*a + 259*cot(c + d*x)**2*csc(c + d*x)**6*sin(c + d*x )**6*b + 252*cot(c + d*x)**2*csc(c + d*x)**6*sin(c + d*x)**5*a + 61*csc(c + d*x)**6*sin(c + d*x)**6*b - 84*csc(c + d*x)**6*sin(c + d*x)**5*a - 189*b )/(5040*sin(c + d*x)**5*d)