3.27 \(\int \frac {\cos (a+b x) \sin ^3(a+b x)}{c+d x} \, dx\)

Optimal. Leaf size=129 \[ -\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Ci}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}-\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d} \]

[Out]

1/4*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d-1/8*cos(4*a-4*b*c/d)*Si(4*b*c/d+4*b*x)/d-1/8*Ci(4*b*c/d+4*b*x)*sin(4*
a-4*b*c/d)/d+1/4*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4406, 3303, 3299, 3302} \[ -\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}-\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(8*d) + (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*
c)/d])/(4*d) + (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(4*d) - (Cos[4*a - (4*b*c)/d]*SinIntegral
[(4*b*c)/d + 4*b*x])/(8*d)

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 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\cos (a+b x) \sin ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac {\sin (2 a+2 b x)}{4 (c+d x)}-\frac {\sin (4 a+4 b x)}{8 (c+d x)}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {\sin (4 a+4 b x)}{c+d x} \, dx\right )+\frac {1}{4} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx\\ &=-\left (\frac {1}{8} \cos \left (4 a-\frac {4 b c}{d}\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx\right )+\frac {1}{4} \cos \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}{8} \sin \left (4 a-\frac {4 b c}{d}\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx+\frac {1}{4} \sin \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 {\text {Ci}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{8 d}+\frac {\text {Ci}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{4 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}-\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 110, normalized size = 0.85 \[ -\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Ci}\left (\frac {4 b (c+d x)}{d}\right )-2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b (c+d x)}{d}\right )-2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] - 2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)
/d] - 2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))
/d])/d

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 156, normalized size = 1.21 \[ \frac {2 \, {\left (\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (\operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {4 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + 4 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{16 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(cos_integral(2*(b*d*x + b*c)/d) + cos_integral(-2*(b*d*x + b*c)/d))*sin(-2*(b*c - a*d)/d) - (cos_inte
gral(4*(b*d*x + b*c)/d) + cos_integral(-4*(b*d*x + b*c)/d))*sin(-4*(b*c - a*d)/d) - 2*cos(-4*(b*c - a*d)/d)*si
n_integral(4*(b*d*x + b*c)/d) + 4*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d))/d

________________________________________________________________________________________

giac [C]  time = 2.11, size = 6046, normalized size = 46.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(
cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c
/d)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 4*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) - 4*real_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))
*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)
^2*tan(2*b*c/d)*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan
(b*c/d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(
-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(4*b*x + 4*b*c/d))*ta
n(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)
^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 - imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*t
an(2*b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2 - 8*imag_part(cos_integra
l(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) + 8*imag_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)*tan(2*b*c/d
)^2*tan(b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 2*imag_part(cos_i
ntegral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2
*a)^2*tan(a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 2*s
in_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2
*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^
2 - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(4
*(b*d*x + b*c)/d)*tan(2*a)*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b
*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_i
ntegral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)
^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - im
ag_part(cos_integral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(2*b*x +
2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b
*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*sin_
integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2*
tan(2*b*c/d)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d) + 2*
real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2*tan(2*b*c/d) - 4*real_part(cos_integral(2*b*x +
2*b*c/d))*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2 - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan
(2*b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2 - 2*real_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*
a)^2*tan(a)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2*tan(b*c/d) + 4*real
_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x -
2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d) - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(2*b*c
/d)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_par
t(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(2*a)^2*tan(a)*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(b*c/d)^2 + 2
*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(4*b*x + 4
*b*c/d))*tan(2*a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b
*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*c
/d))*tan(2*a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^
2*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*real_part(c
os_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*a)^2*tan(a)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)^2 - 2*imag_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(a)^2 - 2*si
n_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(a)^2 + 4*
imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d) - 4*imag_part(cos_integral(-4*b*x - 4*
b*c/d))*tan(2*a)*tan(a)^2*tan(2*b*c/d) + 8*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(a)^2*tan(2*b*c/d) + im
ag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*
tan(2*a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - imag_part(
cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2
*b*c/d)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c
/d))*tan(a)^2*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2 - 2*imag_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(2*b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2
*tan(2*b*c/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(2*b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d
)*tan(a)^2*tan(2*b*c/d)^2 - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d) + 8*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a)*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)
^2*tan(a)*tan(b*c/d) - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) + 8*imag_pa
rt(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(a
)*tan(2*b*c/d)^2*tan(b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 + 2*imag_part(c
os_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2
*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(b*c/d)^2 - 2*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2*tan(b*c/d)^2 + imag_part(cos_in
tegral(4*b*x + 4*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d
)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*
c/d))*tan(a)^2*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 - 4*sin_integral(2*(b*d*
x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)*tan(b*c/d
)^2 - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(4*(b*d*x
+ b*c)/d)*tan(2*a)*tan(2*b*c/d)*tan(b*c/d)^2 - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2*tan(b*
c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*
b*x - 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2*tan(b*c
/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(
2*b*c/d)^2*tan(b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2*tan(a) - 4*real_part(cos_integ
ral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(a) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(a)^2 + 2*re
al_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(a)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a
)^2*tan(2*b*c/d) + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 2*real_part(cos_integ
ral(4*b*x + 4*b*c/d))*tan(a)^2*tan(2*b*c/d) - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2*tan(2*b*c/d
) - 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b
*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(2*b*c/d)^2 - 4*real_par
t(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(2*b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^
2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2*tan(b*c/d) - 4*real_part(cos_integral(2*
b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) + 4*real
_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*t
an(2*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(b*c/d)^2 + 2*real_part(cos_
integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(b*c/d)^2 + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/
d)^2 + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*real_part(cos_integral(4*b*x + 4*b*
c/d))*tan(2*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)*tan(b*c/d)^2 - imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 - 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*a)^2 + 2*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*a)^2 + imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*
sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2 - 4*sin_integral(2*(b*d*x + b*c)/d)*tan(2*a)^2 + imag_part(cos_inte
gral(4*b*x + 4*b*c/d))*tan(a)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - 2*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(a)^2 - imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(a)^2 + 2*sin_integral(4*(b*d*x
+ b*c)/d)*tan(a)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 + 4*imag_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*a)*tan(2*b*c/d) - 4*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 8*sin_integral(4*(b
*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) - imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 2*imag_part(c
os_integral(2*b*x + 2*b*c/d))*tan(2*b*c/d)^2 + 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*b*c/d)^2 + im
ag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 4*
sin_integral(2*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 8*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(b*c/d)
+ 8*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 16*sin_integral(2*(b*d*x + b*c)/d)*tan(a)*ta
n(b*c/d) + imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(b*c/d)^2 + 2*imag_part(cos_integral(2*b*x + 2*b*c/d))*
tan(b*c/d)^2 - 2*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - imag_part(cos_integral(-4*b*x - 4*b*
c/d))*tan(b*c/d)^2 + 2*sin_integral(4*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 4*sin_integral(2*(b*d*x + b*c)/d)*tan(b*
c/d)^2 + 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*a) - 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) - 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)
- 2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2
*b*c/d) + 4*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 4*real_part(cos_integral(-2*b*x - 2*b*c/d))*
tan(b*c/d) + imag_part(cos_integral(4*b*x + 4*b*c/d)) - 2*imag_part(cos_integral(2*b*x + 2*b*c/d)) + 2*imag_pa
rt(cos_integral(-2*b*x - 2*b*c/d)) - imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 2*sin_integral(4*(b*d*x + b*c
)/d) - 4*sin_integral(2*(b*d*x + b*c)/d))/(d*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2*ta
n(a)^2*tan(2*b*c/d)^2 + d*tan(2*a)^2*tan(a)^2*tan(b*c/d)^2 + d*tan(2*a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(
a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2*tan(a)^2 + d*tan(2*a)^2*tan(2*b*c/d)^2 + d*tan(a)^2*tan(2*b*c/
d)^2 + d*tan(2*a)^2*tan(b*c/d)^2 + d*tan(a)^2*tan(b*c/d)^2 + d*tan(2*b*c/d)^2*tan(b*c/d)^2 + d*tan(2*a)^2 + d*
tan(a)^2 + d*tan(2*b*c/d)^2 + d*tan(b*c/d)^2 + d)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 178, normalized size = 1.38 \[ \frac {-\frac {b \left (\frac {4 \Si \left (4 b x +4 a +\frac {-4 d a +4 c b}{d}\right ) \cos \left (\frac {-4 d a +4 c b}{d}\right )}{d}-\frac {4 \Ci \left (4 b x +4 a +\frac {-4 d a +4 c b}{d}\right ) \sin \left (\frac {-4 d a +4 c b}{d}\right )}{d}\right )}{32}+\frac {b \left (\frac {2 \Si \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}-\frac {2 \Ci \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{8}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/32*b*(4*Si(4*b*x+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d-4*Ci(4*b*x+4*a+4*(-a*d+b*c)/d)*sin(4*(-a*d+
b*c)/d)/d)+1/8*b*(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))

________________________________________________________________________________________

maxima [C]  time = 0.47, size = 274, normalized size = 2.12 \[ \frac {b {\left (-2 i \, E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + 2 i \, E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (i \, E_{1}\left (\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b {\left (E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right ) + E_{1}\left (-\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )}{16 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*(-2*I*exp_integral_e(1, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + 2*I*exp_integral_e(1, -(2*I*b*c + 2
*I*(b*x + a)*d - 2*I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b*(I*exp_integral_e(1, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*
a*d)/d) - I*exp_integral_e(1, -(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*cos(-4*(b*c - a*d)/d) - 2*b*(exp_inte
gral_e(1, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + exp_integral_e(1, -(2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/
d))*sin(-2*(b*c - a*d)/d) + b*(exp_integral_e(1, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d) + exp_integral_e(1,
-(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*sin(-4*(b*c - a*d)/d))/(b*d)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________