\(\int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx\) [372]

Optimal result

Integrand size = 23, antiderivative size = 71 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\log (c+d x)}{d}-\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4516, 3393, 3384, 3380, 3383} \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}-\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\log (c+d x)}{d} \]

[In]

Int[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x),x]

[Out]

(2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d + Log[c + d*x]/d - (2*Sin[2*a - (2*b*c)/d]*SinIntegr
al[(2*b*c)/d + 2*b*x])/d

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 3393

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )+\log (c+d x)-2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{d} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.49

method	result	size
risch	\(\frac {\ln \left (d x +c \right )}{d}-\frac {{\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{d}-\frac {{\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{d}\)	\(106\)
default	\(-\frac {\ln \left (d x +c \right )}{d}+\frac {2 \,\operatorname {Si}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {2 \ln \left (-a d +c b +d \left (x b +a \right )\right )}{d}\)	\(116\)

[In]

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

[Out]

ln(d*x+c)/d-1/d*exp(-2*I*(a*d-b*c)/d)*Ei(1,2*I*b*x+2*I*a-2*I*(a*d-b*c)/d)-1/d*exp(2*I*(a*d-b*c)/d)*Ei(1,-2*I*x
*b-2*I*a-2*(-I*a*d+I*c*b)/d)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=-\frac {{\left (E_{1}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \log \left (d x + c\right )}{d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.44 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \, b \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 2 \, b \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + b \log \left (b c + {\left (b x + a\right )} d - a d\right )}{b d} \]

[In]

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

[Out]

(2*b*cos(-2*(b*c - a*d)/d)*cos_integral(2*(b*c + (b*x + a)*d - a*d)/d) + 2*b*sin(-2*(b*c - a*d)/d)*sin_integra
l(-2*(b*c + (b*x + a)*d - a*d)/d) + b*log(b*c + (b*x + a)*d - a*d))/(b*d)

Mupad [F(-1)]

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

[In]

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

[Out]

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