Integrand size = 14, antiderivative size = 78 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {3 \cos ^2(x)}{d (c+d x)}+\frac {4 \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right ) \sin \left (\frac {2 c}{d}\right )}{d^2}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^2} \]
-3*cos(x)^2/d/(d*x+c)-4*cos(2*c/d)*Si(2*c/d+2*x)/d^2+4*Ci(2*c/d+2*x)*sin(2 *c/d)/d^2+sin(x)^2/d/(d*x+c)
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\frac {-\frac {d (1+2 \cos (2 x))}{c+d x}+4 \operatorname {CosIntegral}\left (2 \left (\frac {c}{d}+x\right )\right ) \sin \left (\frac {2 c}{d}\right )-4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (2 \left (\frac {c}{d}+x\right )\right )}{d^2} \]
(-((d*(1 + 2*Cos[2*x]))/(c + d*x)) + 4*CosIntegral[2*(c/d + x)]*Sin[(2*c)/ d] - 4*Cos[(2*c)/d]*SinIntegral[2*(c/d + x)])/d^2
Time = 0.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4931, 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.
\(\displaystyle \int \frac {\sin (3 x) \csc (x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 4931 |
\(\displaystyle \int \left (\frac {3 \cos ^2(x)}{(c+d x)^2}-\frac {\sin ^2(x)}{(c+d x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 \sin \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d^2}-\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^2}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {3 \cos ^2(x)}{d (c+d x)}\) |
(-3*Cos[x]^2)/(d*(c + d*x)) + (4*CosIntegral[(2*c)/d + 2*x]*Sin[(2*c)/d])/ d^2 + Sin[x]^2/(d*(c + d*x)) - (4*Cos[(2*c)/d]*SinIntegral[(2*c)/d + 2*x]) /d^2
3.4.66.3.1 Defintions of rubi rules used
Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x] ^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Member Q[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && E qQ[b*c - a*d, 0] && IGtQ[b/d, 1]
Time = 1.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {2 \cos \left (2 x \right )}{\left (d x +c \right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (\frac {2 c}{d}+2 x \right ) \cos \left (\frac {2 c}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (\frac {2 c}{d}+2 x \right ) \sin \left (\frac {2 c}{d}\right )}{d}\right )}{d}-\frac {1}{d \left (d x +c \right )}\) | \(82\) |
risch | \(-\frac {1}{d \left (d x +c \right )}+\frac {2 i {\mathrm e}^{\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (2 i x +\frac {2 i c}{d}\right )}{d^{2}}-\frac {2 i {\mathrm e}^{-\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (-2 i x -\frac {2 i c}{d}\right )}{d^{2}}-\frac {2 i \cos \left (2 x \right )}{d^{2} \left (i x +\frac {i c}{d}\right )}\) | \(94\) |
-2*cos(2*x)/(d*x+c)/d-2*(2*Si(2*c/d+2*x)*cos(2*c/d)/d-2*Ci(2*c/d+2*x)*sin( 2*c/d)/d)/d-1/d/(d*x+c)
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {4 \, d \cos \left (x\right )^{2} - 4 \, {\left (d x + c\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) + 4 \, {\left (d x + c\right )} \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - d}{d^{3} x + c d^{2}} \]
-(4*d*cos(x)^2 - 4*(d*x + c)*cos_integral(2*(d*x + c)/d)*sin(2*c/d) + 4*(d *x + c)*cos(2*c/d)*sin_integral(2*(d*x + c)/d) - d)/(d^3*x + c*d^2)
\[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\int \frac {\sin {\left (3 x \right )} \csc {\left (x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.23 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {{\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{3} + {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )^{3} + {\left ({\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2\right )} \sin \left (\frac {2 \, c}{d}\right )^{2} + {\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2 \, \cos \left (\frac {2 \, c}{d}\right )^{2} + {\left ({\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{2} - i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )}{2 \, {\left ({\left (\cos \left (\frac {2 \, c}{d}\right )^{2} + \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d^{2} x + {\left (c \cos \left (\frac {2 \, c}{d}\right )^{2} + c \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d\right )}} \]
-1/2*((exp_integral_e(2, 2*(-I*d*x - I*c)/d) + exp_integral_e(2, -2*(-I*d* x - I*c)/d))*cos(2*c/d)^3 + (-I*exp_integral_e(2, 2*(-I*d*x - I*c)/d) + I* exp_integral_e(2, -2*(-I*d*x - I*c)/d))*sin(2*c/d)^3 + ((exp_integral_e(2, 2*(-I*d*x - I*c)/d) + exp_integral_e(2, -2*(-I*d*x - I*c)/d))*cos(2*c/d) + 2)*sin(2*c/d)^2 + (exp_integral_e(2, 2*(-I*d*x - I*c)/d) + exp_integral_ e(2, -2*(-I*d*x - I*c)/d))*cos(2*c/d) + 2*cos(2*c/d)^2 + ((-I*exp_integral _e(2, 2*(-I*d*x - I*c)/d) + I*exp_integral_e(2, -2*(-I*d*x - I*c)/d))*cos( 2*c/d)^2 - I*exp_integral_e(2, 2*(-I*d*x - I*c)/d) + I*exp_integral_e(2, - 2*(-I*d*x - I*c)/d))*sin(2*c/d))/((cos(2*c/d)^2 + sin(2*c/d)^2)*d^2*x + (c *cos(2*c/d)^2 + c*sin(2*c/d)^2)*d)
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.42 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\frac {4 \, d x \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) - 4 \, d x \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 4 \, c \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) - 4 \, c \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - 2 \, d \cos \left (2 \, x\right ) - d}{d^{3} x + c d^{2}} \]
(4*d*x*cos_integral(2*(d*x + c)/d)*sin(2*c/d) - 4*d*x*cos(2*c/d)*sin_integ ral(2*(d*x + c)/d) + 4*c*cos_integral(2*(d*x + c)/d)*sin(2*c/d) - 4*c*cos( 2*c/d)*sin_integral(2*(d*x + c)/d) - 2*d*cos(2*x) - d)/(d^3*x + c*d^2)
Timed out. \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (3\,x\right )}{\sin \left (x\right )\,{\left (c+d\,x\right )}^2} \,d x \]