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

Optimal. Leaf size=57 \[ \frac{2 \cos \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (\frac{2 c}{d}+2 x\right )}{d}+\frac{2 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d}+\frac{\log (c+d x)}{d} \]

[Out]

(2*Cos[(2*c)/d]*CosIntegral[(2*c)/d + 2*x])/d + Log[c + d*x]/d + (2*Sin[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d

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Rubi [A]  time = 0.251852, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {4431, 3312, 3303, 3299, 3302} \[ \frac{2 \cos \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (\frac{2 c}{d}+2 x\right )}{d}+\frac{2 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d}+\frac{\log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x]*Sin[3*x])/(c + d*x),x]

[Out]

(2*Cos[(2*c)/d]*CosIntegral[(2*c)/d + 2*x])/d + Log[c + d*x]/d + (2*Sin[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d

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 3312

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

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} \, dx &=\int \left (\frac{3 \cos ^2(x)}{c+d x}-\frac{\sin ^2(x)}{c+d x}\right ) \, dx\\ &=3 \int \frac{\cos ^2(x)}{c+d x} \, dx-\int \frac{\sin ^2(x)}{c+d x} \, dx\\ &=3 \int \left (\frac{1}{2 (c+d x)}+\frac{\cos (2 x)}{2 (c+d x)}\right ) \, dx-\int \left (\frac{1}{2 (c+d x)}-\frac{\cos (2 x)}{2 (c+d x)}\right ) \, dx\\ &=\frac{\log (c+d x)}{d}+\frac{1}{2} \int \frac{\cos (2 x)}{c+d x} \, dx+\frac{3}{2} \int \frac{\cos (2 x)}{c+d x} \, dx\\ &=\frac{\log (c+d x)}{d}+\frac{1}{2} \cos \left (\frac{2 c}{d}\right ) \int \frac{\cos \left (\frac{2 c}{d}+2 x\right )}{c+d x} \, dx+\frac{1}{2} \left (3 \cos \left (\frac{2 c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c}{d}+2 x\right )}{c+d x} \, dx+\frac{1}{2} \sin \left (\frac{2 c}{d}\right ) \int \frac{\sin \left (\frac{2 c}{d}+2 x\right )}{c+d x} \, dx+\frac{1}{2} \left (3 \sin \left (\frac{2 c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c}{d}+2 x\right )}{c+d x} \, dx\\ &=\frac{2 \cos \left (\frac{2 c}{d}\right ) \text{Ci}\left (\frac{2 c}{d}+2 x\right )}{d}+\frac{\log (c+d x)}{d}+\frac{2 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (\frac{2 c}{d}+2 x\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0631687, size = 49, normalized size = 0.86 \[ \frac{2 \cos \left (\frac{2 c}{d}\right ) \text{CosIntegral}\left (2 \left (\frac{c}{d}+x\right )\right )+2 \sin \left (\frac{2 c}{d}\right ) \text{Si}\left (2 \left (\frac{c}{d}+x\right )\right )+\log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[(2*c)/d]*CosIntegral[2*(c/d + x)] + Log[c + d*x] + 2*Sin[(2*c)/d]*SinIntegral[2*(c/d + x)])/d

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Maple [A]  time = 0.051, size = 58, normalized size = 1. \begin{align*} 2\,{\frac{1}{d}{\it Ci} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \cos \left ( 2\,{\frac{c}{d}} \right ) }+{\frac{\ln \left ( dx+c \right ) }{d}}+2\,{\frac{1}{d}{\it Si} \left ( 2\,{\frac{c}{d}}+2\,x \right ) \sin \left ( 2\,{\frac{c}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Ci(2*c/d+2*x)*cos(2*c/d)/d+ln(d*x+c)/d+2*Si(2*c/d+2*x)*sin(2*c/d)/d

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Maxima [C]  time = 1.23435, size = 128, normalized size = 2.25 \begin{align*} -\frac{{\left (E_{1}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) + E_{1}\left (-\frac{2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac{2 \, c}{d}\right ) -{\left (-i \, E_{1}\left (\frac{2 i \, d x + 2 i \, c}{d}\right ) + i \, E_{1}\left (-\frac{2 i \, d x + 2 i \, c}{d}\right )\right )} \sin \left (\frac{2 \, c}{d}\right ) - \log \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((exp_integral_e(1, (2*I*d*x + 2*I*c)/d) + exp_integral_e(1, -(2*I*d*x + 2*I*c)/d))*cos(2*c/d) - (-I*exp_inte
gral_e(1, (2*I*d*x + 2*I*c)/d) + I*exp_integral_e(1, -(2*I*d*x + 2*I*c)/d))*sin(2*c/d) - log(d*x + c))/d

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Fricas [A]  time = 0.504552, size = 182, normalized size = 3.19 \begin{align*} \frac{{\left (\operatorname{Ci}\left (\frac{2 \,{\left (d x + c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{2 \,{\left (d x + c\right )}}{d}\right )\right )} \cos \left (\frac{2 \, c}{d}\right ) + 2 \, \sin \left (\frac{2 \, c}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((cos_integral(2*(d*x + c)/d) + cos_integral(-2*(d*x + c)/d))*cos(2*c/d) + 2*sin(2*c/d)*sin_integral(2*(d*x +
c)/d) + log(d*x + c))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (3 x \right )} \csc{\left (x \right )}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12076, size = 69, normalized size = 1.21 \begin{align*} \frac{2 \, \cos \left (\frac{2 \, c}{d}\right ) \operatorname{Ci}\left (\frac{2 \,{\left (d x + c\right )}}{d}\right ) + 2 \, \sin \left (\frac{2 \, c}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*cos(2*c/d)*cos_integral(2*(d*x + c)/d) + 2*sin(2*c/d)*sin_integral(2*(d*x + c)/d) + log(d*x + c))/d