Integrand size = 15, antiderivative size = 39 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \cot (c+b x)}{b}-\frac {\csc ^2(c+b x) \sin (a-c)}{2 b} \] Output:
-cos(a-c)*cot(b*x+c)/b-1/2*csc(b*x+c)^2*sin(a-c)/b
Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\frac {(\cos (a)-\cos (a-c) \cos (c+2 b x)) \csc (c) \csc ^2(c+b x)}{2 b} \] Input:
Integrate[Csc[c + b*x]^3*Sin[a + b*x],x]
Output:
((Cos[a] - Cos[a - c]*Cos[c + 2*b*x])*Csc[c]*Csc[c + b*x]^2)/(2*b)
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5093, 3042, 25, 3086, 15, 4254, 24}
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.
\(\displaystyle \int \sin (a+b x) \csc ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 5093 |
\(\displaystyle \cos (a-c) \int \csc ^2(c+b x)dx+\sin (a-c) \int \cot (c+b x) \csc ^2(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos (a-c) \int \csc (c+b x)^2dx+\sin (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right )^2 \tan \left (c+b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \cos (a-c) \int \csc (c+b x)^2dx-\sin (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \cos (a-c) \int \csc (c+b x)^2dx-\frac {\sin (a-c) \int \csc (c+b x)d\csc (c+b x)}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \cos (a-c) \int \csc (c+b x)^2dx-\frac {\sin (a-c) \csc ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\cos (a-c) \int 1d\cot (c+b x)}{b}-\frac {\sin (a-c) \csc ^2(b x+c)}{2 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\cos (a-c) \cot (b x+c)}{b}-\frac {\sin (a-c) \csc ^2(b x+c)}{2 b}\) |
Input:
Int[Csc[c + b*x]^3*Sin[a + b*x],x]
Output:
-((Cos[a - c]*Cot[c + b*x])/b) - (Csc[c + b*x]^2*Sin[a - c])/(2*b)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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]
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62
method | result | size |
risch | \(\frac {i \left (-2 \,{\mathrm e}^{i \left (2 b x +5 a +c \right )}+{\mathrm e}^{i \left (5 a -c \right )}+{\mathrm e}^{i \left (3 a +c \right )}\right )}{\left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}\) | \(63\) |
parallelrisch | \(-\frac {\csc \left (\frac {b x}{2}+\frac {c}{2}\right ) \left (\sin \left (b x +a \right ) \left (-\frac {\sec \left (\frac {b x}{2}+\frac {c}{2}\right )^{2}}{2}+1\right ) \csc \left (\frac {b x}{2}+\frac {c}{2}\right )+\sec \left (\frac {b x}{2}+\frac {c}{2}\right ) \cos \left (b x +a \right )\right )}{4 b}\) | \(63\) |
default | \(\frac {-\frac {\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )}{2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2}}-\frac {1}{\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2} \left (\tan \left (b x +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (b x +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}}{b}\) | \(120\) |
Input:
int(csc(b*x+c)^3*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
I/(-exp(2*I*(b*x+a+c))+exp(2*I*a))^2/b*(-2*exp(I*(2*b*x+5*a+c))+exp(I*(5*a -c))+exp(I*(3*a+c)))
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.21 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) - \sin \left (-a + c\right )}{2 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )}} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a),x, algorithm="fricas")
Output:
1/2*(2*cos(b*x + c)*cos(-a + c)*sin(b*x + c) - sin(-a + c))/(b*cos(b*x + c )^2 - b)
Timed out. \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+c)**3*sin(b*x+a),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (37) = 74\).
Time = 0.05 (sec) , antiderivative size = 399, normalized size of antiderivative = 10.23 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\frac {{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) - {\left (\sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \, \cos \left (a + c\right ) \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) + {\left (\cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (a + c\right ) - 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) - b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) - b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a),x, algorithm="maxima")
Output:
((2*sin(2*b*x + 2*a + 2*c) - sin(2*a) - sin(2*c))*cos(4*b*x + a + 5*c) - 2 *(2*sin(2*b*x + 2*a + 2*c) - sin(2*a) - sin(2*c))*cos(2*b*x + a + 3*c) - ( sin(2*a) + sin(2*c))*cos(a + c) - (2*cos(2*b*x + 2*a + 2*c) - cos(2*a) - c os(2*c))*sin(4*b*x + a + 5*c) + 2*cos(a + c)*sin(2*b*x + 2*a + 2*c) + 2*(2 *cos(2*b*x + 2*a + 2*c) - cos(2*a) - cos(2*c))*sin(2*b*x + a + 3*c) + (cos (2*a) + cos(2*c))*sin(a + c) - 2*cos(2*b*x + 2*a + 2*c)*sin(a + c))/(b*cos (4*b*x + a + 5*c)^2 + 4*b*cos(2*b*x + a + 3*c)^2 - 4*b*cos(2*b*x + a + 3*c )*cos(a + c) + b*cos(a + c)^2 + b*sin(4*b*x + a + 5*c)^2 + 4*b*sin(2*b*x + a + 3*c)^2 - 4*b*sin(2*b*x + a + 3*c)*sin(a + c) + b*sin(a + c)^2 - 2*(2* b*cos(2*b*x + a + 3*c) - b*cos(a + c))*cos(4*b*x + a + 5*c) - 2*(2*b*sin(2 *b*x + a + 3*c) - b*sin(a + c))*sin(4*b*x + a + 5*c))
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (37) = 74\).
Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.72 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=-\frac {\tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + c\right ) + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b \tan \left (b x + c\right )^{2}} \] Input:
integrate(csc(b*x+c)^3*sin(b*x+a),x, algorithm="giac")
Output:
-(tan(b*x + c)*tan(1/2*a)^2*tan(1/2*c)^2 - tan(b*x + c)*tan(1/2*a)^2 + 4*t an(b*x + c)*tan(1/2*a)*tan(1/2*c) + tan(1/2*a)^2*tan(1/2*c) - tan(b*x + c) *tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^2 + tan(b*x + c) + tan(1/2*a) - 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*t an(b*x + c)^2)
Timed out. \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)/sin(c + b*x)^3,x)
Output:
\text{Hanged}
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08 \[ \int \csc ^3(c+b x) \sin (a+b x) \, dx=\frac {-\cos \left (b x +c \right ) \sin \left (b x +a \right )-\cos \left (b x +a \right ) \sin \left (b x +c \right )}{2 \sin \left (b x +c \right )^{2} b} \] Input:
int(csc(b*x+c)^3*sin(b*x+a),x)
Output:
( - (cos(b*x + c)*sin(a + b*x) + cos(a + b*x)*sin(b*x + c)))/(2*sin(b*x + c)**2*b)