3.199 \(\int \csc ^5(c+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=60 \[ -\frac{\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^4(b x+c)}{4 b} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0466843, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4582, 2606, 30, 3767} \[ -\frac{\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^4(b x+c)}{4 b} \]

Antiderivative was successfully verified.

[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 4582

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]

Rule 2606

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 30

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

Rule 3767

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]

Rubi steps

\begin{align*} \int \csc ^5(c+b x) \sin (a+b x) \, dx &=\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) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+b x)\right )}{b}-\frac{\sin (a-c) \operatorname{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]  time = 0.374245, size = 58, normalized size = 0.97 \[ \frac{\csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^4(b x+c) (\cos (a-c) (\cos (4 b x+3 c)-4 \cos (2 b x+c))+3 \cos (a))}{24 b} \]

Antiderivative was successfully verified.

[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 [B]  time = 1.888, size = 321, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.57422, size = 1453, normalized size = 24.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [A]  time = 0.481041, size = 193, normalized size = 3.22 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.15743, size = 406, normalized size = 6.77 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)