\(\int \csc ^6(c+b x) \sin (a+b x) \, dx\) [86]

Optimal result

Integrand size = 15, antiderivative size = 94 \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (c+b x)) \cos (a-c)}{8 b}-\frac {3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac {\csc ^5(c+b x) \sin (a-c)}{5 b} \] Output:

-3/8*arctanh(cos(b*x+c))*cos(a-c)/b-3/8*cos(a-c)*cot(b*x+c)*csc(b*x+c)/b-1 
/4*cos(a-c)*cot(b*x+c)*csc(b*x+c)^3/b-1/5*csc(b*x+c)^5*sin(a-c)/b
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {480 \text {arctanh}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+2 \csc ^5(c+b x) (64 \sin (a-c)+5 \cos (a-c) (14 \sin (2 (c+b x))-3 \sin (4 (c+b x))))}{640 b} \] Input:

Integrate[Csc[c + b*x]^6*Sin[a + b*x],x]
 

Output:

-1/640*(480*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 2*Csc[c + b 
*x]^5*(64*Sin[a - c] + 5*Cos[a - c]*(14*Sin[2*(c + b*x)] - 3*Sin[4*(c + b* 
x)])))/b
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5093, 3042, 25, 3086, 15, 4255, 3042, 4255, 3042, 4257}

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.

\(\displaystyle \int \sin (a+b x) \csc ^6(b x+c) \, dx\)

\(\Big \downarrow \) 5093

\(\displaystyle \cos (a-c) \int \csc ^5(c+b x)dx+\sin (a-c) \int \cot (c+b x) \csc ^5(c+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos (a-c) \int \csc (c+b x)^5dx+\sin (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right )^5 \tan \left (c+b x-\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle \cos (a-c) \int \csc (c+b x)^5dx-\sin (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right )^5 \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx\)

\(\Big \downarrow \) 3086

\(\displaystyle \cos (a-c) \int \csc (c+b x)^5dx-\frac {\sin (a-c) \int \csc ^4(c+b x)d\csc (c+b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \cos (a-c) \int \csc (c+b x)^5dx-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

\(\Big \downarrow \) 4255

\(\displaystyle \cos (a-c) \left (\frac {3}{4} \int \csc ^3(c+b x)dx-\frac {\cot (b x+c) \csc ^3(b x+c)}{4 b}\right )-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos (a-c) \left (\frac {3}{4} \int \csc (c+b x)^3dx-\frac {\cot (b x+c) \csc ^3(b x+c)}{4 b}\right )-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

\(\Big \downarrow \) 4255

\(\displaystyle \cos (a-c) \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cot (b x+c) \csc ^3(b x+c)}{4 b}\right )-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \cos (a-c) \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+b x)dx-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cot (b x+c) \csc ^3(b x+c)}{4 b}\right )-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

\(\Big \downarrow \) 4257

\(\displaystyle \cos (a-c) \left (\frac {3}{4} \left (-\frac {\text {arctanh}(\cos (b x+c))}{2 b}-\frac {\cot (b x+c) \csc (b x+c)}{2 b}\right )-\frac {\cot (b x+c) \csc ^3(b x+c)}{4 b}\right )-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}\)

Input:

Int[Csc[c + b*x]^6*Sin[a + b*x],x]
 

Output:

Cos[a - c]*(-1/4*(Cot[c + b*x]*Csc[c + b*x]^3)/b + (3*(-1/2*ArcTanh[Cos[c 
+ b*x]]/b - (Cot[c + b*x]*Csc[c + b*x])/(2*b)))/4) - (Csc[c + b*x]^5*Sin[a 
 - c])/(5*b)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 5093

Int[Csc[w_]^(n_.)*Sin[v_], x_Symbol] :> Simp[Sin[v - w]   Int[Cot[w]*Csc[w] 
^(n - 1), x], x] + Simp[Cos[v - w]   Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0 
] && FreeQ[v - w, x] && NeQ[w, v]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 19.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.73

method	result	size
risch	\(\frac {-15 \,{\mathrm e}^{i \left (9 b x +11 a +8 c \right )}-15 \,{\mathrm e}^{i \left (9 b x +9 a +10 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +11 a +6 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +9 a +8 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +11 a +4 c \right )}-128 \,{\mathrm e}^{i \left (5 b x +9 a +6 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +11 a +2 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +9 a +4 c \right )}+15 \,{\mathrm e}^{i \left (b x +11 a \right )}+15 \,{\mathrm e}^{i \left (b x +9 a +2 c \right )}}{40 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(6746\)

Input:

int(csc(b*x+c)^6*sin(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/40/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^5*(-15*exp(I*(9*b*x+11*a+8*c))-15* 
exp(I*(9*b*x+9*a+10*c))+70*exp(I*(7*b*x+11*a+6*c))+70*exp(I*(7*b*x+9*a+8*c 
))+128*exp(I*(5*b*x+11*a+4*c))-128*exp(I*(5*b*x+9*a+6*c))-70*exp(I*(3*b*x+ 
11*a+2*c))-70*exp(I*(3*b*x+9*a+4*c))+15*exp(I*(b*x+11*a))+15*exp(I*(b*x+9* 
a+2*c)))+3/8*ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*cos(a-c)-3/8*ln(exp(I*(b*x+ 
a))+exp(I*(a-c)))/b*cos(a-c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (86) = 172\).

Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.10 \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=-\frac {15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 10 \, {\left (3 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 5 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) - 16 \, \sin \left (-a + c\right )}{80 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )} \sin \left (b x + c\right )} \] Input:

integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="fricas")
 

Output:

-1/80*(15*(cos(b*x + c)^4*cos(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos 
(-a + c))*log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c) - 15*(cos(b*x + c)^4*co 
s(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos(-a + c))*log(-1/2*cos(b*x + 
 c) + 1/2)*sin(b*x + c) - 10*(3*cos(b*x + c)^3*cos(-a + c) - 5*cos(b*x + c 
)*cos(-a + c))*sin(b*x + c) - 16*sin(-a + c))/((b*cos(b*x + c)^4 - 2*b*cos 
(b*x + c)^2 + b)*sin(b*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Timed out} \] Input:

integrate(csc(b*x+c)**6*sin(b*x+a),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3879 vs. \(2 (86) = 172\).

Time = 0.21 (sec) , antiderivative size = 3879, normalized size of antiderivative = 41.27 \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="maxima")
 

Output:

1/80*(2*(15*cos(9*b*x + 2*a + 8*c) + 15*cos(9*b*x + 10*c) - 70*cos(7*b*x + 
 2*a + 6*c) - 70*cos(7*b*x + 8*c) - 128*cos(5*b*x + 2*a + 4*c) + 128*cos(5 
*b*x + 6*c) + 70*cos(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(b*x 
 + 2*a) - 15*cos(b*x + 2*c))*cos(10*b*x + a + 10*c) - 30*(5*cos(8*b*x + a 
+ 8*c) - 10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + 
 a + 2*c) + cos(a))*cos(9*b*x + 2*a + 8*c) - 30*(5*cos(8*b*x + a + 8*c) - 
10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) 
 + cos(a))*cos(9*b*x + 10*c) + 10*(70*cos(7*b*x + 2*a + 6*c) + 70*cos(7*b* 
x + 8*c) + 128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*cos(3*b* 
x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*cos(b*x + 2*a) + 15*cos(b*x + 2* 
c))*cos(8*b*x + a + 8*c) - 140*(10*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a 
 + 4*c) + 5*cos(2*b*x + a + 2*c) - cos(a))*cos(7*b*x + 2*a + 6*c) - 140*(1 
0*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) 
- cos(a))*cos(7*b*x + 8*c) - 20*(128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b* 
x + 6*c) - 70*cos(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*cos(b*x + 
2*a) + 15*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) + 256*(10*cos(4*b*x + a + 4 
*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*x + 2*a + 4*c) - 256*(10*co 
s(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*x + 6*c) - 1 
00*(14*cos(3*b*x + 2*a + 2*c) + 14*cos(3*b*x + 4*c) - 3*cos(b*x + 2*a) - 3 
*cos(b*x + 2*c))*cos(4*b*x + a + 4*c) + 140*(5*cos(2*b*x + a + 2*c) - c...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8035 vs. \(2 (86) = 172\).

Time = 0.19 (sec) , antiderivative size = 8035, normalized size of antiderivative = 85.48 \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="giac")
 

Output:

1/320*(120*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/ 
2*c) - tan(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x + 1/2*c)))/(tan(1/2*a)^2*tan( 
1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (4*tan(1/2*b*x + 1/2*c)^5*ta 
n(1/2*a)^10*tan(1/2*c)^9 - 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c 
)^10 - 5*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^10 + 16*tan(1/2*b 
*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^7 - 12*tan(1/2*b*x + 1/2*c)^5*tan(1 
/2*a)^9*tan(1/2*c)^8 - 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^ 
8 + 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1/2*c)^9 - 20*tan(1/2*b*x + 
 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^9 + 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) 
^10*tan(1/2*c)^9 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^10 - 
15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/2*c)^10 - 20*tan(1/2*b*x + 1/ 
2*c)^3*tan(1/2*a)^9*tan(1/2*c)^10 - 40*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^1 
0*tan(1/2*c)^10 + 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^5 - 8 
*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c)^6 - 10*tan(1/2*b*x + 1/2*c 
)^4*tan(1/2*a)^10*tan(1/2*c)^6 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*ta 
n(1/2*c)^7 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^7 + 80*tan( 
1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(1/2*c)^7 - 48*tan(1/2*b*x + 1/2*c)^5* 
tan(1/2*a)^7*tan(1/2*c)^8 - 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/2 
*c)^8 - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^8 - 120*tan(1/2* 
b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2*c)^8 + 8*tan(1/2*b*x + 1/2*c)^5*ta...
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=\text {Hanged} \] Input:

int(sin(a + b*x)/sin(c + b*x)^6,x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \csc ^6(c+b x) \sin (a+b x) \, dx=\frac {-28 \cos \left (b x +c \right ) \cos \left (b x +a \right ) \sin \left (b x +c \right )^{3}+24 \cos \left (b x +c \right ) \cos \left (b x +a \right ) \sin \left (b x +c \right )-36 \cos \left (b x +c \right ) \sin \left (b x +c \right )^{4} \sin \left (b x +a \right )-24 \cos \left (b x +c \right ) \sin \left (b x +c \right )^{3}-48 \cos \left (b x +c \right ) \sin \left (b x +c \right )^{2} \sin \left (b x +a \right )+24 \cos \left (b x +c \right ) \sin \left (b x +c \right )-32 \cos \left (b x +c \right ) \sin \left (b x +a \right )-60 \cos \left (b x +a \right ) \sin \left (b x +c \right )^{3}+16 \cos \left (b x +a \right ) \sin \left (b x +c \right )-60 \left (\int \frac {\tan \left (\frac {b x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}d x \right ) \sin \left (b x +c \right )^{5} b +12 \left (\int \frac {1}{\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{5} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{5}}d x \right ) \sin \left (b x +c \right )^{5} b +36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (b x +c \right )^{5}-12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {c}{2}\right )\right ) \sin \left (b x +c \right )^{5}+33 \sin \left (b x +c \right )^{5}+28 \sin \left (b x +c \right )^{4} \sin \left (b x +a \right )-36 \sin \left (b x +c \right )^{3}-8 \sin \left (b x +c \right )^{2} \sin \left (b x +a \right )+24 \sin \left (b x +c \right )}{160 \sin \left (b x +c \right )^{5} b} \] Input:

int(csc(b*x+c)^6*sin(b*x+a),x)
 

Output:

( - 28*cos(b*x + c)*cos(a + b*x)*sin(b*x + c)**3 + 24*cos(b*x + c)*cos(a + 
 b*x)*sin(b*x + c) - 36*cos(b*x + c)*sin(b*x + c)**4*sin(a + b*x) - 24*cos 
(b*x + c)*sin(b*x + c)**3 - 48*cos(b*x + c)*sin(b*x + c)**2*sin(a + b*x) + 
 24*cos(b*x + c)*sin(b*x + c) - 32*cos(b*x + c)*sin(a + b*x) - 60*cos(a + 
b*x)*sin(b*x + c)**3 + 16*cos(a + b*x)*sin(b*x + c) - 60*int(tan((b*x + c) 
/2)/(tan((a + b*x)/2)**2 + 1),x)*sin(b*x + c)**5*b + 12*int(1/(tan((b*x + 
c)/2)**5*tan((a + b*x)/2)**2 + tan((b*x + c)/2)**5),x)*sin(b*x + c)**5*b + 
 36*log(tan((b*x + c)/2)**2 + 1)*sin(b*x + c)**5 - 12*log(tan((b*x + c)/2) 
)*sin(b*x + c)**5 + 33*sin(b*x + c)**5 + 28*sin(b*x + c)**4*sin(a + b*x) - 
 36*sin(b*x + c)**3 - 8*sin(b*x + c)**2*sin(a + b*x) + 24*sin(b*x + c))/(1 
60*sin(b*x + c)**5*b)