\(\int \csc ^5(c+b x) \sin (a+b x) \, dx\) [199]

Optimal result

Integrand size = 15, antiderivative size = 60 \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \cot (c+b x)}{b}-\frac {\cos (a-c) \cot ^3(c+b x)}{3 b}-\frac {\csc ^4(c+b x) \sin (a-c)}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4678, 2686, 30, 3852} \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c)}{b}-\frac {\sin (a-c) \csc ^4(b x+c)}{4 b} \]

[In]

Int[Csc[c + b*x]^5*Sin[a + b*x],x]

[Out]

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

Rule 30

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

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4678

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

Rubi steps \begin{align*} \text {integral}& = \cos (a-c) \int \csc ^4(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^4(c+b x) \, dx \\ & = -\frac {\cos (a-c) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+b x)\right )}{b}-\frac {\sin (a-c) \text {Subst}\left (\int x^3 \, dx,x,\csc (c+b x)\right )}{b} \\ & = -\frac {\cos (a-c) \cot (c+b x)}{b}-\frac {\cos (a-c) \cot ^3(c+b x)}{3 b}-\frac {\csc ^4(c+b x) \sin (a-c)}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=\frac {(3 \cos (a)+\cos (a-c) (-4 \cos (c+2 b x)+\cos (3 c+4 b x))) \csc \left (\frac {c}{2}\right ) \csc ^4(c+b x) \sec \left (\frac {c}{2}\right )}{24 b} \]

[In]

Integrate[Csc[c + b*x]^5*Sin[a + b*x],x]

[Out]

((3*Cos[a] + Cos[a - c]*(-4*Cos[c + 2*b*x] + Cos[3*c + 4*b*x]))*Csc[c/2]*Csc[c + b*x]^4*Sec[c/2])/(24*b)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.75 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {2 i \left (6 \,{\mathrm e}^{i \left (4 x b +9 a +3 c \right )}-4 \,{\mathrm e}^{i \left (2 x b +9 a +c \right )}-4 \,{\mathrm e}^{i \left (2 x b +7 a +3 c \right )}+{\mathrm e}^{i \left (9 a -c \right )}+{\mathrm e}^{i \left (7 a +c \right )}\right )}{3 \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )^{4} b}\)	\(97\)
parallelrisch	\(-\frac {\left (\sec \left (\frac {x b}{2}+\frac {c}{2}\right ) \left (\sin \left (2 x b +a +c \right )-\frac {\sin \left (4 x b +a +3 c \right )}{4}+\sin \left (a -c \right )-\frac {\sin \left (-2 x b +a -3 c \right )}{4}\right ) \csc \left (\frac {x b}{2}+\frac {c}{2}\right )+3 \cos \left (x b +a \right )-\cos \left (-x b +a -2 c \right )-\cos \left (3 x b +a +2 c \right )\right ) \csc \left (\frac {x b}{2}+\frac {c}{2}\right )^{3} \sec \left (\frac {x b}{2}+\frac {c}{2}\right )^{3}}{96 b}\)	\(119\)
default	\(\frac {-\frac {3 \sin \left (a \right ) \cos \left (c \right )-3 \cos \left (a \right ) \sin \left (c \right )}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {1}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}-\frac {\cos \left (a \right )^{2} \cos \left (c \right )^{2}+3 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+3 \cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}{3 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{3}}-\frac {\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right )}{4 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{4} \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{4}}}{b}\)	\(321\)

[In]

int(csc(b*x+c)^5*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2/3*I/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^4/b*(6*exp(I*(4*b*x+9*a+3*c))-4*exp(I*(2*b*x+9*a+c))-4*exp(I*(2*b*x+7*a
+3*c))+exp(I*(9*a-c))+exp(I*(7*a+c)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25 \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=\frac {4 \, {\left (2 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) + 3 \, \sin \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )}} \]

[In]

integrate(csc(b*x+c)^5*sin(b*x+a),x, algorithm="fricas")

[Out]

1/12*(4*(2*cos(b*x + c)^3*cos(-a + c) - 3*cos(b*x + c)*cos(-a + c))*sin(b*x + c) + 3*sin(-a + c))/(b*cos(b*x +
c)^4 - 2*b*cos(b*x + c)^2 + b)

Sympy [F(-1)]

Timed out. \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+c)**5*sin(b*x+a),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (56) = 112\).

Time = 0.24 (sec) , antiderivative size = 1076, normalized size of antiderivative = 17.93 \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate(csc(b*x+c)^5*sin(b*x+a),x, algorithm="maxima")

[Out]

-2/3*((6*sin(4*b*x + 2*a + 4*c) - 4*sin(2*b*x + 2*a + 2*c) - 4*sin(2*b*x + 4*c) + sin(2*a) + sin(2*c))*cos(8*b
*x + a + 9*c) - 4*(6*sin(4*b*x + 2*a + 4*c) - 4*sin(2*b*x + 2*a + 2*c) - 4*sin(2*b*x + 4*c) + sin(2*a) + sin(2
*c))*cos(6*b*x + a + 7*c) + 6*(4*sin(2*b*x + a + 3*c) - sin(a + c))*cos(4*b*x + 2*a + 4*c) + 6*(6*sin(4*b*x +
2*a + 4*c) - 4*sin(2*b*x + 2*a + 2*c) - 4*sin(2*b*x + 4*c) + sin(2*a) + sin(2*c))*cos(4*b*x + a + 5*c) + 4*(4*
sin(2*b*x + 2*a + 2*c) - sin(2*a) - sin(2*c))*cos(2*b*x + a + 3*c) - 4*(4*sin(2*b*x + a + 3*c) - sin(a + c))*c
os(2*b*x + 4*c) + (sin(2*a) + sin(2*c))*cos(a + c) - (6*cos(4*b*x + 2*a + 4*c) - 4*cos(2*b*x + 2*a + 2*c) - 4*
cos(2*b*x + 4*c) + cos(2*a) + cos(2*c))*sin(8*b*x + a + 9*c) + 4*(6*cos(4*b*x + 2*a + 4*c) - 4*cos(2*b*x + 2*a
+ 2*c) - 4*cos(2*b*x + 4*c) + cos(2*a) + cos(2*c))*sin(6*b*x + a + 7*c) - 6*(4*cos(2*b*x + a + 3*c) - cos(a +
c))*sin(4*b*x + 2*a + 4*c) - 6*(6*cos(4*b*x + 2*a + 4*c) - 4*cos(2*b*x + 2*a + 2*c) - 4*cos(2*b*x + 4*c) + co
s(2*a) + cos(2*c))*sin(4*b*x + a + 5*c) - 4*cos(a + c)*sin(2*b*x + 2*a + 2*c) - 4*(4*cos(2*b*x + 2*a + 2*c) -
cos(2*a) - cos(2*c))*sin(2*b*x + a + 3*c) + 4*(4*cos(2*b*x + a + 3*c) - cos(a + c))*sin(2*b*x + 4*c) - (cos(2*
a) + cos(2*c))*sin(a + c) + 4*cos(2*b*x + 2*a + 2*c)*sin(a + c))/(b*cos(8*b*x + a + 9*c)^2 + 16*b*cos(6*b*x +
a + 7*c)^2 + 36*b*cos(4*b*x + a + 5*c)^2 + 16*b*cos(2*b*x + a + 3*c)^2 - 8*b*cos(2*b*x + a + 3*c)*cos(a + c) +
b*cos(a + c)^2 + b*sin(8*b*x + a + 9*c)^2 + 16*b*sin(6*b*x + a + 7*c)^2 + 36*b*sin(4*b*x + a + 5*c)^2 + 16*b*
sin(2*b*x + a + 3*c)^2 - 8*b*sin(2*b*x + a + 3*c)*sin(a + c) + b*sin(a + c)^2 - 2*(4*b*cos(6*b*x + a + 7*c) -
6*b*cos(4*b*x + a + 5*c) + 4*b*cos(2*b*x + a + 3*c) - b*cos(a + c))*cos(8*b*x + a + 9*c) - 8*(6*b*cos(4*b*x +
a + 5*c) - 4*b*cos(2*b*x + a + 3*c) + b*cos(a + c))*cos(6*b*x + a + 7*c) - 12*(4*b*cos(2*b*x + a + 3*c) - b*co
s(a + c))*cos(4*b*x + a + 5*c) - 2*(4*b*sin(6*b*x + a + 7*c) - 6*b*sin(4*b*x + a + 5*c) + 4*b*sin(2*b*x + a +
3*c) - b*sin(a + c))*sin(8*b*x + a + 9*c) - 8*(6*b*sin(4*b*x + a + 5*c) - 4*b*sin(2*b*x + a + 3*c) + b*sin(a +
c))*sin(6*b*x + a + 7*c) - 12*(4*b*sin(2*b*x + a + 3*c) - b*sin(a + c))*sin(4*b*x + a + 5*c))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (56) = 112\).

Time = 0.32 (sec) , antiderivative size = 301, normalized size of antiderivative = 5.02 \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=-\frac {6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} + 24 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (b x + c\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (b x + c\right )^{3} + 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 6 \, \tan \left (b x + c\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 8 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (b x + c\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right ) - 3 \, \tan \left (\frac {1}{2} \, c\right )}{6 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b \tan \left (b x + c\right )^{4}} \]

[In]

integrate(csc(b*x+c)^5*sin(b*x+a),x, algorithm="giac")

[Out]

-1/6*(6*tan(b*x + c)^3*tan(1/2*a)^2*tan(1/2*c)^2 - 6*tan(b*x + c)^3*tan(1/2*a)^2 + 24*tan(b*x + c)^3*tan(1/2*a
)*tan(1/2*c) + 6*tan(b*x + c)^2*tan(1/2*a)^2*tan(1/2*c) - 6*tan(b*x + c)^3*tan(1/2*c)^2 - 6*tan(b*x + c)^2*tan
(1/2*a)*tan(1/2*c)^2 + 2*tan(b*x + c)*tan(1/2*a)^2*tan(1/2*c)^2 + 6*tan(b*x + c)^3 + 6*tan(b*x + c)^2*tan(1/2*
a) - 2*tan(b*x + c)*tan(1/2*a)^2 - 6*tan(b*x + c)^2*tan(1/2*c) + 8*tan(b*x + c)*tan(1/2*a)*tan(1/2*c) + 3*tan(
1/2*a)^2*tan(1/2*c) - 2*tan(b*x + c)*tan(1/2*c)^2 - 3*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(b*x + c) + 3*tan(1/2*a)
- 3*tan(1/2*c))/((tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*b*tan(b*x + c)^4)

Mupad [F(-1)]

Timed out. \[ \int \csc ^5(c+b x) \sin (a+b x) \, dx=\text {Hanged} \]

[In]

int(sin(a + b*x)/sin(c + b*x)^5,x)

[Out]

\text{Hanged}