3.373 \(\int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx\)
Optimal. Leaf size=102 \[ -\frac {4 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {3 \cos ^2(a+b x)}{d (c+d x)} \]
[Out]
-3*cos(b*x+a)^2/d/(d*x+c)-4*b*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d^2-4*b*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^
2+sin(b*x+a)^2/d/(d*x+c)
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Rubi [A] time = 0.28, antiderivative size = 102, normalized size of antiderivative = 1.00,
number of steps used = 12, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used
= {4431, 3313, 12, 3303, 3299, 3302} \[ -\frac {4 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}-\frac {4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {3 \cos ^2(a+b x)}{d (c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^2,x]
[Out]
(-3*Cos[a + b*x]^2)/(d*(c + d*x)) - (4*b*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^2 + Sin[a + b*
x]^2/(d*(c + d*x)) - (4*b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*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 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]
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 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 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]
Rubi steps
\begin {align*} \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx &=\int \left (\frac {3 \cos ^2(a+b x)}{(c+d x)^2}-\frac {\sin ^2(a+b x)}{(c+d x)^2}\right ) \, dx\\ &=3 \int \frac {\cos ^2(a+b x)}{(c+d x)^2} \, dx-\int \frac {\sin ^2(a+b x)}{(c+d x)^2} \, dx\\ &=-\frac {3 \cos ^2(a+b x)}{d (c+d x)}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {(2 b) \int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx}{d}+\frac {(6 b) \int -\frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx}{d}\\ &=-\frac {3 \cos ^2(a+b x)}{d (c+d x)}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {b \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx}{d}-\frac {(3 b) \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx}{d}\\ &=-\frac {3 \cos ^2(a+b x)}{d (c+d x)}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {\left (b \cos \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}{d}-\frac {\left (3 b \cos \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}{d}-\frac {\left (b \sin \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}{d}-\frac {\left (3 b \sin \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}{d}\\ &=-\frac {3 \cos ^2(a+b x)}{d (c+d x)}-\frac {4 b \text {Ci}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{d^2}+\frac {\sin ^2(a+b x)}{d (c+d x)}-\frac {4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 81, normalized size = 0.79 \[ -\frac {4 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b (c+d x)}{d}\right )+4 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+\frac {d (2 \cos (2 (a+b x))+1)}{c+d x}}{d^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^2,x]
[Out]
-(((d*(1 + 2*Cos[2*(a + b*x)]))/(c + d*x) + 4*b*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + 4*b*Cos[
2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^2)
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fricas [A] time = 0.45, size = 131, normalized size = 1.28 \[ -\frac {4 \, d \cos \left (b x + a\right )^{2} + 4 \, {\left (b d x + b c\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + 2 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d}{d^{3} x + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^2,x, algorithm="fricas")
[Out]
-(4*d*cos(b*x + a)^2 + 4*(b*d*x + b*c)*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) + 2*((b*d*x + b*c
)*cos_integral(2*(b*d*x + b*c)/d) + (b*d*x + b*c)*cos_integral(-2*(b*d*x + b*c)/d))*sin(-2*(b*c - a*d)/d) - d)
/(d^3*x + c*d^2)
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giac [C] time = 10.37, size = 5381, normalized size = 52.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^2,x, algorithm="giac")
[Out]
(2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*ta
n(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*
tan(b*c/d) + 4*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) - 8*b*d*x*re
al_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 - 8*b*d*x*real_part(cos_integral(-
2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*
x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b
*c/d)^2 + 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_i
ntegral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4 + 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)
^2*tan(1/2*a)^4 - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^4 + 16*b*d*x*imag_part(cos_int
egral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) - 16*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) + 32*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3*tan(b*
c/d) + 4*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) + 4*b*c*real_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d) - 12*b*d*x*imag_part(cos_integral(2*b*x + 2*b*
c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 12*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan
(1/2*a)^2*tan(b*c/d)^2 - 24*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 8*b*c
*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 - 8*b*c*real_part(cos_integral(
-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d)^2 + 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan
(1/2*a)^4*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b*d*x
*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^4*tan(b*c/d)^2 + 8*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*
tan(b*x)^2*tan(1/2*a)^3 + 8*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3 - 2*b*c*im
ag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^4 + 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d
))*tan(b*x)^2*tan(1/2*a)^4 - 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^4 - 24*b*d*x*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) - 24*b*d*x*real_part(cos_integral(-2*b*x -
2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 16*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan
(1/2*a)^3*tan(b*c/d) - 16*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) + 3
2*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^3*tan(b*c/d) + 4*b*d*x*real_part(cos_integral(2*b*
x + 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) + 4*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c
/d) + 8*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 + 8*b*d*x*real_part(
cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 12*b*c*imag_part(cos_integral(2*b*x + 2*b
*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + 12*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(
1/2*a)^2*tan(b*c/d)^2 - 24*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 8*b*d*x*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 - 8*b*d*x*real_part(cos_integral(-2*b*x - 2
*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 + 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2
- 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d)^2 + 4*b*c*sin_integral(2*(b*d*x + b*
c)/d)*tan(1/2*a)^4*tan(b*c/d)^2 + d*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + 12*b*d*x*imag_part(cos_integral(2*b
*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 - 12*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2
*a)^2 + 24*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)^2 + 8*b*c*real_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3 + 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^3
- 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4 + 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*
c/d))*tan(1/2*a)^4 - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^4 - 16*b*d*x*imag_part(cos_integral(2*
b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) + 16*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)
^2*tan(1/2*a)*tan(b*c/d) - 32*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 24*b*c*
real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) - 24*b*c*real_part(cos_integral(-2
*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d) + 16*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/
2*a)^3*tan(b*c/d) - 16*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 32*b*d*x*sin_
integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3*tan(b*c/d) + 4*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a
)^4*tan(b*c/d) + 4*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4*tan(b*c/d) + 2*b*d*x*imag_part(c
os_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(
b*x)^2*tan(b*c/d)^2 + 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)^2 + 8*b*c*real_part(cos_in
tegral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 + 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(b*x)^2*tan(1/2*a)*tan(b*c/d)^2 - 12*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)
^2 + 12*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b*d*x*sin_integral(2*(b
*d*x + b*c)/d)*tan(1/2*a)^2*tan(b*c/d)^2 - 8*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c
/d)^2 - 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d)^2 - 8*b*d*x*real_part(cos_inte
gral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a) - 8*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*ta
n(1/2*a) + 12*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 - 12*b*c*imag_part(cos_inte
gral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)^2 + 24*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)
^2 + 8*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3 + 8*b*d*x*real_part(cos_integral(-2*b*x - 2
*b*c/d))*tan(1/2*a)^3 - 2*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^4 + 2*b*c*imag_part(cos_inte
gral(-2*b*x - 2*b*c/d))*tan(1/2*a)^4 - 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^4 + d*tan(b*x)^2*tan(1
/2*a)^4 + 4*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 4*b*d*x*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 16*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(
1/2*a)*tan(b*c/d) + 16*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 32*b*c
*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(1/2*a)*tan(b*c/d) - 24*b*d*x*real_part(cos_integral(2*b*x + 2*
b*c/d))*tan(1/2*a)^2*tan(b*c/d) - 24*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) +
16*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) - 16*b*c*imag_part(cos_integral(-2*b*
x - 2*b*c/d))*tan(1/2*a)^3*tan(b*c/d) + 32*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^3*tan(b*c/d) + 2*b*c
*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*
c/d))*tan(b*x)^2*tan(b*c/d)^2 + 4*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(b*c/d)^2 + 8*b*d*x*real_p
art(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 + 8*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))
*tan(1/2*a)*tan(b*c/d)^2 - 12*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 + 12*b*c*
imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d)^2 - 24*b*c*sin_integral(2*(b*d*x + b*c)/d)*t
an(1/2*a)^2*tan(b*c/d)^2 - 14*d*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 - 16*d*tan(b*x)*tan(1/2*a)^3*tan(b*c/d)^2
- 3*d*tan(1/2*a)^4*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2 + 2*b*d*x*imag_
part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2 - 8*b*c*r
eal_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(1/2*a) - 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(b*x)^2*tan(1/2*a) + 12*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2 - 12*b*d*x*imag_part(
cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2 + 24*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^2 + 8*b*c*r
eal_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^3 + 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2
*a)^3 + 4*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(b*c/d) + 4*b*c*real_part(cos_integral(-2
*b*x - 2*b*c/d))*tan(b*x)^2*tan(b*c/d) - 16*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/
d) + 16*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d) - 32*b*d*x*sin_integral(2*(b*d*x
+ b*c)/d)*tan(1/2*a)*tan(b*c/d) - 24*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) - 2
4*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2*tan(b*c/d) + 2*b*d*x*imag_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 4*b*d*x*sin_integ
ral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 8*b*c*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 +
8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(b*x)^2 + 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2 - 4*b*c*sin_integral(2*(b*
d*x + b*c)/d)*tan(b*x)^2 - 8*b*d*x*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a) - 8*b*d*x*real_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(1/2*a) + 12*b*c*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)^2 - 12*b*
c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a)^2 + 24*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)^2
- 14*d*tan(b*x)^2*tan(1/2*a)^2 - 16*d*tan(b*x)*tan(1/2*a)^3 - 3*d*tan(1/2*a)^4 + 4*b*d*x*real_part(cos_integr
al(2*b*x + 2*b*c/d))*tan(b*c/d) + 4*b*d*x*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) - 16*b*c*imag_p
art(cos_integral(2*b*x + 2*b*c/d))*tan(1/2*a)*tan(b*c/d) + 16*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*ta
n(1/2*a)*tan(b*c/d) - 32*b*c*sin_integral(2*(b*d*x + b*c)/d)*tan(1/2*a)*tan(b*c/d) + 2*b*c*imag_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(b*c/d)^2 - 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 + 4*b*c*sin_
integral(2*(b*d*x + b*c)/d)*tan(b*c/d)^2 + d*tan(b*x)^2*tan(b*c/d)^2 + 16*d*tan(b*x)*tan(1/2*a)*tan(b*c/d)^2 +
10*d*tan(1/2*a)^2*tan(b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/d)) + 2*b*d*x*imag_part(cos_int
egral(-2*b*x - 2*b*c/d)) - 4*b*d*x*sin_integral(2*(b*d*x + b*c)/d) - 8*b*c*real_part(cos_integral(2*b*x + 2*b*
c/d))*tan(1/2*a) - 8*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(1/2*a) + 4*b*c*real_part(cos_integral(2
*b*x + 2*b*c/d))*tan(b*c/d) + 4*b*c*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d) - 2*b*c*imag_part(cos
_integral(2*b*x + 2*b*c/d)) + 2*b*c*imag_part(cos_integral(-2*b*x - 2*b*c/d)) - 4*b*c*sin_integral(2*(b*d*x +
b*c)/d) + d*tan(b*x)^2 + 16*d*tan(b*x)*tan(1/2*a) + 10*d*tan(1/2*a)^2 - 3*d*tan(b*c/d)^2 - 3*d)/(d^3*x*tan(b*x
)^2*tan(1/2*a)^4*tan(b*c/d)^2 + c*d^2*tan(b*x)^2*tan(1/2*a)^4*tan(b*c/d)^2 + d^3*x*tan(b*x)^2*tan(1/2*a)^4 + 2
*d^3*x*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + d^3*x*tan(1/2*a)^4*tan(b*c/d)^2 + c*d^2*tan(b*x)^2*tan(1/2*a)^4
+ 2*c*d^2*tan(b*x)^2*tan(1/2*a)^2*tan(b*c/d)^2 + c*d^2*tan(1/2*a)^4*tan(b*c/d)^2 + 2*d^3*x*tan(b*x)^2*tan(1/2*
a)^2 + d^3*x*tan(1/2*a)^4 + d^3*x*tan(b*x)^2*tan(b*c/d)^2 + 2*d^3*x*tan(1/2*a)^2*tan(b*c/d)^2 + 2*c*d^2*tan(b*
x)^2*tan(1/2*a)^2 + c*d^2*tan(1/2*a)^4 + c*d^2*tan(b*x)^2*tan(b*c/d)^2 + 2*c*d^2*tan(1/2*a)^2*tan(b*c/d)^2 + d
^3*x*tan(b*x)^2 + 2*d^3*x*tan(1/2*a)^2 + d^3*x*tan(b*c/d)^2 + c*d^2*tan(b*x)^2 + 2*c*d^2*tan(1/2*a)^2 + c*d^2*
tan(b*c/d)^2 + d^3*x + c*d^2)
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maple [A] time = 0.04, size = 169, normalized size = 1.66 \[ \frac {1}{d \left (d x +c \right )}+\frac {b^{2} \left (-\frac {2 \cos \left (2 b x +2 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {2 \left (\frac {2 \Si \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}-\frac {2 \Ci \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{d}\right )-\frac {2 b^{2}}{\left (\left (b x +a \right ) d -d a +c b \right ) d}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^2,x)
[Out]
1/d/(d*x+c)+4/b*(1/4*b^2*(-2*cos(2*b*x+2*a)/((b*x+a)*d-d*a+c*b)/d-2*(2*Si(2*b*x+2*a+2*(-a*d+b*c)/d)*cos(2*(-a*
d+b*c)/d)/d-2*Ci(2*b*x+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d)/d)-1/2*b^2/((b*x+a)*d-d*a+c*b)/d)
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maxima [C] time = 0.42, size = 118, normalized size = 1.16 \[ -\frac {{\left (E_{2}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) + E_{2}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (i \, E_{2}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) - i \, E_{2}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 1}{d^{2} x + c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)^2,x, algorithm="maxima")
[Out]
-((exp_integral_e(2, (2*I*b*d*x + 2*I*b*c)/d) + exp_integral_e(2, -(2*I*b*d*x + 2*I*b*c)/d))*cos(-2*(b*c - a*d
)/d) - (I*exp_integral_e(2, (2*I*b*d*x + 2*I*b*c)/d) - I*exp_integral_e(2, -(2*I*b*d*x + 2*I*b*c)/d))*sin(-2*(
b*c - a*d)/d) + 1)/(d^2*x + c*d)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (3\,a+3\,b\,x\right )}{\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(3*a + 3*b*x)/(sin(a + b*x)*(c + d*x)^2),x)
[Out]
int(sin(3*a + 3*b*x)/(sin(a + b*x)*(c + d*x)^2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(3*b*x+3*a)/(d*x+c)**2,x)
[Out]
Timed out
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