Integrand size = 29, antiderivative size = 138 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {b^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 d}+\frac {4 a b \csc ^5(c+d x)}{5 d}+\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-\frac {a^2 \csc ^8(c+d x)}{8 d} \] Output:
-1/2*b^2*csc(d*x+c)^2/d-2/3*a*b*csc(d*x+c)^3/d-1/4*(a^2-2*b^2)*csc(d*x+c)^ 4/d+4/5*a*b*csc(d*x+c)^5/d+1/6*(2*a^2-b^2)*csc(d*x+c)^6/d-2/7*a*b*csc(d*x+ c)^7/d-1/8*a^2*csc(d*x+c)^8/d
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\csc ^2(c+d x) \left (420 b^2+560 a b \csc (c+d x)+210 \left (a^2-2 b^2\right ) \csc ^2(c+d x)-672 a b \csc ^3(c+d x)-140 \left (2 a^2-b^2\right ) \csc ^4(c+d x)+240 a b \csc ^5(c+d x)+105 a^2 \csc ^6(c+d x)\right )}{840 d} \] Input:
Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
Output:
-1/840*(Csc[c + d*x]^2*(420*b^2 + 560*a*b*Csc[c + d*x] + 210*(a^2 - 2*b^2) *Csc[c + d*x]^2 - 672*a*b*Csc[c + d*x]^3 - 140*(2*a^2 - b^2)*Csc[c + d*x]^ 4 + 240*a*b*Csc[c + d*x]^5 + 105*a^2*Csc[c + d*x]^6))/d
Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
| \(\displaystyle \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (a+b \sin (c+d x))^2}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \csc ^9(c+d x) (a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^2d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^4 \int \frac {\csc ^9(c+d x) (a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^9}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {b^4 \int \left (\frac {a^2 \csc ^9(c+d x)}{b^5}+\frac {2 a \csc ^8(c+d x)}{b^4}+\frac {\left (b^4-2 a^2 b^2\right ) \csc ^7(c+d x)}{b^7}-\frac {4 a \csc ^6(c+d x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \csc ^5(c+d x)}{b^5}+\frac {2 a \csc ^4(c+d x)}{b^4}+\frac {\csc ^3(c+d x)}{b^3}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^4 \left (-\frac {a^2 \csc ^8(c+d x)}{8 b^4}+\frac {\left (2 a^2-b^2\right ) \csc ^6(c+d x)}{6 b^4}-\frac {\left (a^2-2 b^2\right ) \csc ^4(c+d x)}{4 b^4}-\frac {2 a \csc ^7(c+d x)}{7 b^3}+\frac {4 a \csc ^5(c+d x)}{5 b^3}-\frac {2 a \csc ^3(c+d x)}{3 b^3}-\frac {\csc ^2(c+d x)}{2 b^2}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
Output:
(b^4*(-1/2*Csc[c + d*x]^2/b^2 - (2*a*Csc[c + d*x]^3)/(3*b^3) - ((a^2 - 2*b ^2)*Csc[c + d*x]^4)/(4*b^4) + (4*a*Csc[c + d*x]^5)/(5*b^3) + ((2*a^2 - b^2 )*Csc[c + d*x]^6)/(6*b^4) - (2*a*Csc[c + d*x]^7)/(7*b^3) - (a^2*Csc[c + d* x]^8)/(8*b^4)))/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 = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(-\frac {\frac {a^{2} \csc \left (d x +c \right )^{8}}{8}+\frac {2 a b \csc \left (d x +c \right )^{7}}{7}+\frac {\left (-2 a^{2}+b^{2}\right ) \csc \left (d x +c \right )^{6}}{6}-\frac {4 a b \csc \left (d x +c \right )^{5}}{5}+\frac {\left (a^{2}-2 b^{2}\right ) \csc \left (d x +c \right )^{4}}{4}+\frac {2 a b \csc \left (d x +c \right )^{3}}{3}+\frac {\csc \left (d x +c \right )^{2} b^{2}}{2}}{d}\) | \(107\) |
| default | \(-\frac {\frac {a^{2} \csc \left (d x +c \right )^{8}}{8}+\frac {2 a b \csc \left (d x +c \right )^{7}}{7}+\frac {\left (-2 a^{2}+b^{2}\right ) \csc \left (d x +c \right )^{6}}{6}-\frac {4 a b \csc \left (d x +c \right )^{5}}{5}+\frac {\left (a^{2}-2 b^{2}\right ) \csc \left (d x +c \right )^{4}}{4}+\frac {2 a b \csc \left (d x +c \right )^{3}}{3}+\frac {\csc \left (d x +c \right )^{2} b^{2}}{2}}{d}\) | \(107\) |
| risch | \(\frac {2 b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-4 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-\frac {16 i {\mathrm e}^{3 i \left (d x +c \right )} a b}{3}-\frac {16 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}}{3}+\frac {26 b^{2} {\mathrm e}^{10 i \left (d x +c \right )}}{3}-\frac {1376 i {\mathrm e}^{7 i \left (d x +c \right )} a b}{105}-\frac {40 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}}{3}-\frac {40 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}}{3}+\frac {16 i b a \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}-\frac {16 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {26 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {1376 i a b \,{\mathrm e}^{9 i \left (d x +c \right )}}{105}-4 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+\frac {16 i a b \,{\mathrm e}^{13 i \left (d x +c \right )}}{3}+2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-\frac {16 i a b \,{\mathrm e}^{11 i \left (d x +c \right )}}{15}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\) | \(272\) |
Input:
int(cot(d*x+c)^5*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/d*(1/8*a^2*csc(d*x+c)^8+2/7*a*b*csc(d*x+c)^7+1/6*(-2*a^2+b^2)*csc(d*x+c )^6-4/5*a*b*csc(d*x+c)^5+1/4*(a^2-2*b^2)*csc(d*x+c)^4+2/3*a*b*csc(d*x+c)^3 +1/2*csc(d*x+c)^2*b^2)
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {420 \, b^{2} \cos \left (d x + c\right )^{6} - 210 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 140 \, b^{2} - 16 \, {\left (35 \, a b \cos \left (d x + c\right )^{4} - 28 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\right )} \sin \left (d x + c\right )}{840 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/840*(420*b^2*cos(d*x + c)^6 - 210*(a^2 + 4*b^2)*cos(d*x + c)^4 + 140*(a^ 2 + 4*b^2)*cos(d*x + c)^2 - 35*a^2 - 140*b^2 - 16*(35*a*b*cos(d*x + c)^4 - 28*a*b*cos(d*x + c)^2 + 8*a*b)*sin(d*x + c))/(d*cos(d*x + c)^8 - 4*d*cos( d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**5*csc(d*x+c)**4*(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} - 672 \, a b \sin \left (d x + c\right )^{3} + 210 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 240 \, a b \sin \left (d x + c\right ) - 140 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/840*(420*b^2*sin(d*x + c)^6 + 560*a*b*sin(d*x + c)^5 - 672*a*b*sin(d*x + c)^3 + 210*(a^2 - 2*b^2)*sin(d*x + c)^4 + 240*a*b*sin(d*x + c) - 140*(2* a^2 - b^2)*sin(d*x + c)^2 + 105*a^2)/(d*sin(d*x + c)^8)
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {420 \, b^{2} \sin \left (d x + c\right )^{6} + 560 \, a b \sin \left (d x + c\right )^{5} + 210 \, a^{2} \sin \left (d x + c\right )^{4} - 420 \, b^{2} \sin \left (d x + c\right )^{4} - 672 \, a b \sin \left (d x + c\right )^{3} - 280 \, a^{2} \sin \left (d x + c\right )^{2} + 140 \, b^{2} \sin \left (d x + c\right )^{2} + 240 \, a b \sin \left (d x + c\right ) + 105 \, a^{2}}{840 \, d \sin \left (d x + c\right )^{8}} \] Input:
integrate(cot(d*x+c)^5*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/840*(420*b^2*sin(d*x + c)^6 + 560*a*b*sin(d*x + c)^5 + 210*a^2*sin(d*x + c)^4 - 420*b^2*sin(d*x + c)^4 - 672*a*b*sin(d*x + c)^3 - 280*a^2*sin(d*x + c)^2 + 140*b^2*sin(d*x + c)^2 + 240*a*b*sin(d*x + c) + 105*a^2)/(d*sin( d*x + c)^8)
Time = 22.76 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\frac {a^2}{8}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{6}\right )+\frac {b^2\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {2\,a\,b\,\sin \left (c+d\,x\right )}{7}-\frac {4\,a\,b\,{\sin \left (c+d\,x\right )}^3}{5}+\frac {2\,a\,b\,{\sin \left (c+d\,x\right )}^5}{3}}{d\,{\sin \left (c+d\,x\right )}^8} \] Input:
int((cot(c + d*x)^5*(a + b*sin(c + d*x))^2)/sin(c + d*x)^4,x)
Output:
-(a^2/8 + sin(c + d*x)^4*(a^2/4 - b^2/2) - sin(c + d*x)^2*(a^2/3 - b^2/6) + (b^2*sin(c + d*x)^6)/2 + (2*a*b*sin(c + d*x))/7 - (4*a*b*sin(c + d*x)^3) /5 + (2*a*b*sin(c + d*x)^5)/3)/(d*sin(c + d*x)^8)
Time = 0.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04 \[ \int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {2555 \sin \left (d x +c \right )^{8} a^{2}+12320 \sin \left (d x +c \right )^{8} b^{2}-53760 \sin \left (d x +c \right )^{6} b^{2}-71680 \sin \left (d x +c \right )^{5} a b -26880 \sin \left (d x +c \right )^{4} a^{2}+53760 \sin \left (d x +c \right )^{4} b^{2}+86016 \sin \left (d x +c \right )^{3} a b +35840 \sin \left (d x +c \right )^{2} a^{2}-17920 \sin \left (d x +c \right )^{2} b^{2}-30720 \sin \left (d x +c \right ) a b -13440 a^{2}}{107520 \sin \left (d x +c \right )^{8} d} \] Input:
int(cot(d*x+c)^5*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x)
Output:
(2555*sin(c + d*x)**8*a**2 + 12320*sin(c + d*x)**8*b**2 - 53760*sin(c + d* x)**6*b**2 - 71680*sin(c + d*x)**5*a*b - 26880*sin(c + d*x)**4*a**2 + 5376 0*sin(c + d*x)**4*b**2 + 86016*sin(c + d*x)**3*a*b + 35840*sin(c + d*x)**2 *a**2 - 17920*sin(c + d*x)**2*b**2 - 30720*sin(c + d*x)*a*b - 13440*a**2)/ (107520*sin(c + d*x)**8*d)