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

Optimal. Leaf size=78 \[ \frac{4 \sin \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (\frac{2 c}{d}+2 x\right )}{d^2}-\frac{4 \cos \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d^2}+\frac{\sin ^2(x)}{d (c+d x)}-\frac{3 \cos ^2(x)}{d (c+d x)} \]

[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

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Rubi [A]  time = 0.237926, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4431, 3313, 12, 3303, 3299, 3302} \[ \frac{4 \sin \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (\frac{2 c}{d}+2 x\right )}{d^2}-\frac{4 \cos \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d^2}+\frac{\sin ^2(x)}{d (c+d x)}-\frac{3 \cos ^2(x)}{d (c+d x)} \]

Antiderivative was successfully verified.

[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 4431

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]

Rule 3313

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 12

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

Rule 3303

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 3299

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 3302

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]

Rubi steps

\begin{align*} \int \frac{\csc (x) \sin (3 x)}{(c+d x)^2} \, dx &=\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 \text{Ci}\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]  time = 0.138473, size = 61, normalized size = 0.78 \[ \frac{4 \sin \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (2 \left (\frac{c}{d}+x\right )\right )-4 \cos \left (\frac{2 c}{d}\right ) \text{Si}\left (2 \left (\frac{c}{d}+x\right )\right )-\frac{d (2 \cos (2 x)+1)}{c+d x}}{d^2} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.054, size = 82, normalized size = 1.1 \begin{align*} -2\,{\frac{\cos \left ( 2\,x \right ) }{ \left ( dx+c \right ) d}}-2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \cos \left ( 2\,{\frac{c}{d}} \right ) }-2\,{\frac{1}{d}{\it Ci} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \sin \left ( 2\,{\frac{c}{d}} \right ) } \right ) }-{\frac{1}{ \left ( dx+c \right ) d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [C]  time = 1.22548, size = 437, normalized size = 5.6 \begin{align*} -\frac{{\left (E_{2}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) + E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac{2 \, c}{d}\right )^{3} +{\left (i \, E_{2}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) - i \, E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{d}\right )\right )} \sin \left (\frac{2 \, c}{d}\right )^{3} +{\left ({\left (E_{2}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) + E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{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 i \, d x + 2 i \, c}{d}\right ) + E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{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 i \, d x + 2 i \, c}{d}\right ) - i \, E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac{2 \, c}{d}\right )^{2} + i \, E_{2}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) - i \, E_{2}\left (-\frac{2 i \, d x + 2 i \, c}{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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 2*I*c)/d) + exp_integral_e(2, -(2*I*d*x + 2*I*c)/d))*cos(2*c/d)^3 + (I*exp
_integral_e(2, (2*I*d*x + 2*I*c)/d) - I*exp_integral_e(2, -(2*I*d*x + 2*I*c)/d))*sin(2*c/d)^3 + ((exp_integral
_e(2, (2*I*d*x + 2*I*c)/d) + exp_integral_e(2, -(2*I*d*x + 2*I*c)/d))*cos(2*c/d) + 2)*sin(2*c/d)^2 + (exp_inte
gral_e(2, (2*I*d*x + 2*I*c)/d) + exp_integral_e(2, -(2*I*d*x + 2*I*c)/d))*cos(2*c/d) + 2*cos(2*c/d)^2 + ((I*ex
p_integral_e(2, (2*I*d*x + 2*I*c)/d) - I*exp_integral_e(2, -(2*I*d*x + 2*I*c)/d))*cos(2*c/d)^2 + I*exp_integra
l_e(2, (2*I*d*x + 2*I*c)/d) - I*exp_integral_e(2, -(2*I*d*x + 2*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)

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Fricas [A]  time = 0.514176, size = 251, normalized size = 3.22 \begin{align*} -\frac{4 \, d \cos \left (x\right )^{2} + 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 ) - 2 \,{\left ({\left (d x + c\right )} \operatorname{Ci}\left (\frac{2 \,{\left (d x + c\right )}}{d}\right ) +{\left (d x + c\right )} \operatorname{Ci}\left (-\frac{2 \,{\left (d x + c\right )}}{d}\right )\right )} \sin \left (\frac{2 \, c}{d}\right ) - d}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(2*c/d)*sin_integral(2*(d*x + c)/d) - 2*((d*x + c)*cos_integral(2*(d*x + c)/d)
+ (d*x + c)*cos_integral(-2*(d*x + c)/d))*sin(2*c/d) - d)/(d^3*x + c*d^2)

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

[Out]

Timed out

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Giac [A]  time = 1.11966, size = 150, normalized size = 1.92 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)