\(\int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx\) [366]

Optimal result

Integrand size = 14, antiderivative size = 78 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {3 \cos ^2(x)}{d (c+d x)}+\frac {4 \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right ) \sin \left (\frac {2 c}{d}\right )}{d^2}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^2} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4516, 3394, 12, 3384, 3380, 3383} \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\frac {4 \sin \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d^2}-\frac {4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d^2}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {3 \cos ^2(x)}{d (c+d x)} \]

[In]

Int[(Csc[x]*Sin[3*x])/(c + d*x)^2,x]

[Out]

(-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*Co
s[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 4516

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] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos ^2(x)}{(c+d x)^2}-\frac {\sin ^2(x)}{(c+d x)^2}\right ) \, dx \\ & = 3 \int \frac {\cos ^2(x)}{(c+d x)^2} \, dx-\int \frac {\sin ^2(x)}{(c+d x)^2} \, dx \\ & = -\frac {3 \cos ^2(x)}{d (c+d x)}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {2 \int \frac {\sin (2 x)}{2 (c+d x)} \, dx}{d}+\frac {6 \int -\frac {\sin (2 x)}{2 (c+d x)} \, dx}{d} \\ & = -\frac {3 \cos ^2(x)}{d (c+d x)}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {\int \frac {\sin (2 x)}{c+d x} \, dx}{d}-\frac {3 \int \frac {\sin (2 x)}{c+d x} \, dx}{d} \\ & = -\frac {3 \cos ^2(x)}{d (c+d x)}+\frac {\sin ^2(x)}{d (c+d x)}-\frac {\cos \left (\frac {2 c}{d}\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d}-\frac {\left (3 \cos \left (\frac {2 c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d}+\frac {\sin \left (\frac {2 c}{d}\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d}+\frac {\left (3 \sin \left (\frac {2 c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx}{d} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\frac {-\frac {d (1+2 \cos (2 x))}{c+d x}+4 \operatorname {CosIntegral}\left (2 \left (\frac {c}{d}+x\right )\right ) \sin \left (\frac {2 c}{d}\right )-4 \cos \left (\frac {2 c}{d}\right ) \text {Si}\left (2 \left (\frac {c}{d}+x\right )\right )}{d^2} \]

[In]

Integrate[(Csc[x]*Sin[3*x])/(c + d*x)^2,x]

[Out]

(-((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

Maple [A] (verified)

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	\(-\frac {1}{d \left (d x +c \right )}+\frac {2 i {\mathrm e}^{\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (2 i x +\frac {2 i c}{d}\right )}{d^{2}}-\frac {2 i {\mathrm e}^{-\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (-2 i x -\frac {2 i c}{d}\right )}{d^{2}}-\frac {2 i \cos \left (2 x \right )}{d^{2} \left (i x +\frac {i c}{d}\right )}\)	\(94\)

[In]

int(csc(x)*sin(3*x)/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {4 \, d \cos \left (x\right )^{2} - 4 \, {\left (d x + c\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) + 4 \, {\left (d x + c\right )} \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - d}{d^{3} x + c d^{2}} \]

[In]

integrate(csc(x)*sin(3*x)/(d*x+c)^2,x, algorithm="fricas")

[Out]

-(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)

Sympy [F]

\[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\int \frac {\sin {\left (3 x \right )} \csc {\left (x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(csc(x)*sin(3*x)/(d*x+c)**2,x)

[Out]

Integral(sin(3*x)*csc(x)/(c + d*x)**2, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.23 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=-\frac {{\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{3} + {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )^{3} + {\left ({\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2\right )} \sin \left (\frac {2 \, c}{d}\right )^{2} + {\left (E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2 \, \cos \left (\frac {2 \, c}{d}\right )^{2} + {\left ({\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right )^{2} - i \, E_{2}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right )}{2 \, {\left ({\left (\cos \left (\frac {2 \, c}{d}\right )^{2} + \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d^{2} x + {\left (c \cos \left (\frac {2 \, c}{d}\right )^{2} + c \sin \left (\frac {2 \, c}{d}\right )^{2}\right )} d\right )}} \]

[In]

integrate(csc(x)*sin(3*x)/(d*x+c)^2,x, algorithm="maxima")

[Out]

-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_inte
gral_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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.42 \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\frac {4 \, d x \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) - 4 \, d x \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 4 \, c \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) \sin \left (\frac {2 \, c}{d}\right ) - 4 \, c \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) - 2 \, d \cos \left (2 \, x\right ) - d}{d^{3} x + c d^{2}} \]

[In]

integrate(csc(x)*sin(3*x)/(d*x+c)^2,x, algorithm="giac")

[Out]

(4*d*x*cos_integral(2*(d*x + c)/d)*sin(2*c/d) - 4*d*x*cos(2*c/d)*sin_integral(2*(d*x + c)/d) + 4*c*cos_integra
l(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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (x) \sin (3 x)}{(c+d x)^2} \, dx=\int \frac {\sin \left (3\,x\right )}{\sin \left (x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(sin(3*x)/(sin(x)*(c + d*x)^2),x)

[Out]

int(sin(3*x)/(sin(x)*(c + d*x)^2), x)