\(\int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 121 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d}+\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \]

[Out]

1/4*cos(a-b*c/d)*Si(b*c/d+b*x)/d+1/4*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d+1/4*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/
d)/d+1/4*Ci(b*c/d+b*x)*sin(a-b*c/d)/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4491, 3384, 3380, 3383} \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \]

[In]

Int[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x),x]

[Out]

(CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4
*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*
x])/(4*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 4491

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*} \text {integral}& = \int \left (\frac {\sin (a+b x)}{4 (c+d x)}+\frac {\sin (3 a+3 b x)}{4 (c+d x)}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\sin (a+b x)}{c+d x} \, dx+\frac {1}{4} \int \frac {\sin (3 a+3 b x)}{c+d x} \, dx \\ & = \frac {1}{4} \cos \left (3 a-\frac {3 b c}{d}\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac {1}{4} \cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx+\frac {1}{4} \sin \left (3 a-\frac {3 b c}{d}\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac {1}{4} \sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx \\ & = \frac {\operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d}+\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+\operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )}{4 d} \]

[In]

Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x),x]

[Out]

(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*
c)/d]*SinIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(4*d)

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(\frac {\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{12}+\frac {b \left (-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{4}}{b}\)	\(172\)
default	\(\frac {\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{12}+\frac {b \left (-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{4}}{b}\)	\(172\)
risch	\(-\frac {i {\mathrm e}^{-\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 i b x +3 i a -\frac {3 i \left (a d -c b \right )}{d}\right )}{8 d}-\frac {i {\mathrm e}^{-\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (i b x +i a -\frac {i \left (a d -c b \right )}{d}\right )}{8 d}+\frac {i {\mathrm e}^{\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-i b x -i a -\frac {-i a d +i c b}{d}\right )}{8 d}+\frac {i {\mathrm e}^{\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-3 i b x -3 i a -\frac {3 \left (-i a d +i c b \right )}{d}\right )}{8 d}\)	\(194\)

[In]

int(cos(b*x+a)^2*sin(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

1/b*(1/12*b*(-3*Si(-3*x*b-3*a-3*(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d-3*Ci(3*x*b+3*a+3*(-a*d+b*c)/d)*sin(3*(-a*d
+b*c)/d)/d)+1/4*b*(-Si(-x*b-a-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci(x*b+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{4 \, d} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

1/4*(cos_integral((b*d*x + b*c)/d)*sin(-(b*c - a*d)/d) + cos_integral(3*(b*d*x + b*c)/d)*sin(-3*(b*c - a*d)/d)
+ cos(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) + cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d))/
d

Sympy [F]

\[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

integrate(cos(b*x+a)**2*sin(b*x+a)/(d*x+c),x)

[Out]

Integral(sin(a + b*x)*cos(a + b*x)**2/(c + d*x), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=-\frac {b {\left (i \, E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b {\left (-i \, E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b {\left (E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, b d} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-1/8*(b*(I*exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp_integral_e(1, -(I*b*c + I*(b*x + a)*d
- I*a*d)/d))*cos(-(b*c - a*d)/d) + b*(-I*exp_integral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integ
ral_e(1, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + b*(exp_integral_e(1, (I*b*c + I*(b*x
+ a)*d - I*a*d)/d) + exp_integral_e(1, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b*(exp_integ
ral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin
(-3*(b*c - a*d)/d))/(b*d)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.51 (sec) , antiderivative size = 6279, normalized size of antiderivative = 51.89 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\text {Too large to display} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

1/8*(imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + im
ag_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos
_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(
-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x + b*c
)/d)*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^
2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2
*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(
3/2*b*c/d)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c
/d)*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)*ta
n(1/2*b*c/d)^2 - 2*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d
)^2 - 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*re
al_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*real_part
(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + imag_part(cos_int
egral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 - imag_part(cos_integral(b*x + b*c/d))*tan(
3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3
/2*b*c/d)^2 - imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + 2*sin_int
egral(3*(b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*
a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + 4*imag_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*
c/d)^2*tan(1/2*b*c/d) - 4*imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1
/2*b*c/d) + 8*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - imag_par
t(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(b*x + b*c
/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a
)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 -
2*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*sin_integral((b*d*x + b*c)/d)
*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 4*imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(1/2*a)
^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 4*imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3
/2*b*c/d)*tan(1/2*b*c/d)^2 + 8*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)*tan(1/2*
b*c/d)^2 + imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part
(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(-b*x - b*c
/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*t
an(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c
/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integra
l(3*b*x + 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(b*x + b*c/d))*tan(
1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)
^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2
- 2*sin_integral(3*(b*d*x + b*c)/d)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_integral((b*d*x +
b*c)/d)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)
^2*tan(1/2*a)^2*tan(3/2*b*c/d) + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2
*b*c/d) + 2*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2 + 2*real_part(cos_in
tegral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2 - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*ta
n(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)^2 - 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*t
an(3/2*b*c/d)^2 + 2*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*real_par
t(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d) - 2*real_part(cos_integral(b*x + b*c/d)
)*tan(3/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(3/2*
b*c/d)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)
+ 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 2*real_part(cos_integ
ral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a
)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(1/2*b
*c/d)^2 + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*real_part(c
os_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(-3*b*x -
3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(1/2*a
)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)*
tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2
*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*real_part(cos_inte
gral(b*x + b*c/d))*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(-b*x - b*c/d))*tan(
1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2
- imag_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(3
/2*a)^2*tan(1/2*a)^2 + imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2 - 2*sin_integral(3*
(b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)^2 + 4*ima
g_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d) - 4*imag_part(cos_integral(-3*b*x
- 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d) + 8*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)*tan(1/2*a)^
2*tan(3/2*b*c/d) + imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2 + imag_part(cos_inte
gral(b*x + b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(3/2*
b*c/d)^2 - imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x +
b*c)/d)*tan(3/2*a)^2*tan(3/2*b*c/d)^2 + 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(3/2*b*c/d)^2 - imag_
part(cos_integral(3*b*x + 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2 - imag_part(cos_integral(b*x + b*c/d))*tan(1
/2*a)^2*tan(3/2*b*c/d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + imag_part(cos
_integral(-3*b*x - 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d)^2 - 2*sin_integral(3*(b*d*x + b*c)/d)*tan(1/2*a)^2*ta
n(3/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + 4*imag_part(cos_integral(b*x
+ b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(1/2*b*c/d) - 4*imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/
2*a)*tan(1/2*b*c/d) + 8*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)*tan(1/2*b*c/d) + 4*imag_part(cos
_integral(b*x + b*c/d))*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 4*imag_part(cos_integral(-b*x - b*c/d))*t
an(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) + 8*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/
2*b*c/d) - imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(b*x
+ b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d)^2
+ imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d)^2 - 2*sin_integral(3*(b*d*x + b*c)/d)
*tan(3/2*a)^2*tan(1/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*b*c/d)^2 + imag_part(cos
_integral(3*b*x + 3*b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*
tan(1/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integra
l(-3*b*x - 3*b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x + b*c)/d)*tan(1/2*a)^2*tan(1/2*b*
c/d)^2 + 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 4*imag_part(cos_integral(3*b*x + 3*b*
c/d))*tan(3/2*a)*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 4*imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(
3/2*b*c/d)*tan(1/2*b*c/d)^2 + 8*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - i
mag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_part(cos_integral(b*x + b*c/d
))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2
+ imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*sin_integral(3*(b*d*x + b*c
)/d)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2
*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a) + 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/
2*a)^2*tan(1/2*a) + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2 + 2*real_part(cos_integ
ral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2 + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(3
/2*b*c/d) + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan(3/2*b*c/d) - 2*real_part(cos_integral
(3*b*x + 3*b*c/d))*tan(1/2*a)^2*tan(3/2*b*c/d) - 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(1/2*a)^2*tan(
3/2*b*c/d) - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d)^2 - 2*real_part(cos_integral
(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d)^2 + 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)*tan(3/2*b*
c/d)^2 + 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)*tan(3/2*b*c/d)^2 - 2*real_part(cos_integral(b*x +
b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d) - 2*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*b*c/d) + 2*
real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(-b*x - b*c/d))*tan
(1/2*a)^2*tan(1/2*b*c/d) - 2*real_part(cos_integral(b*x + b*c/d))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 2*real_par
t(cos_integral(-b*x - b*c/d))*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan
(3/2*a)*tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*b*c/d)^2 - 2*real_pa
rt(cos_integral(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)
*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 2*real_part(c
os_integral(-3*b*x - 3*b*c/d))*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(
3/2*a)^2 + imag_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*a
)^2 + imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2 - 2*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)^2
+ 2*sin_integral((b*d*x + b*c)/d)*tan(3/2*a)^2 + imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(1/2*a)^2 - imag_
part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2 - imag_part(
cos_integral(-3*b*x - 3*b*c/d))*tan(1/2*a)^2 + 2*sin_integral(3*(b*d*x + b*c)/d)*tan(1/2*a)^2 - 2*sin_integral
((b*d*x + b*c)/d)*tan(1/2*a)^2 + 4*imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d) - 4*imag
_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(3/2*b*c/d) + 8*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*a)
*tan(3/2*b*c/d) - imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*b*c/d)^2 + imag_part(cos_integral(b*x + b*c
/d))*tan(3/2*b*c/d)^2 - imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*b*c/d)^2 + imag_part(cos_integral(-3*b*x
- 3*b*c/d))*tan(3/2*b*c/d)^2 - 2*sin_integral(3*(b*d*x + b*c)/d)*tan(3/2*b*c/d)^2 + 2*sin_integral((b*d*x + b
*c)/d)*tan(3/2*b*c/d)^2 + 4*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) - 4*imag_part(cos_i
ntegral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) + 8*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)*tan(1/2*b*c/d) +
imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(1/2*b*c/d)^2 - imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*c/
d)^2 + imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/d)^2 - imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(
1/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x + b*c)/d)*tan(1/2*b*c/d)^2 - 2*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*
c/d)^2 + 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a) + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*t
an(3/2*a) + 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a) + 2*real_part(cos_integral(-b*x - b*c/d))*tan(1/
2*a) - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*b*c/d) - 2*real_part(cos_integral(-3*b*x - 3*b*c/d))
*tan(3/2*b*c/d) - 2*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d) - 2*real_part(cos_integral(-b*x - b*c/
d))*tan(1/2*b*c/d) + imag_part(cos_integral(3*b*x + 3*b*c/d)) + imag_part(cos_integral(b*x + b*c/d)) - imag_pa
rt(cos_integral(-b*x - b*c/d)) - imag_part(cos_integral(-3*b*x - 3*b*c/d)) + 2*sin_integral(3*(b*d*x + b*c)/d)
+ 2*sin_integral((b*d*x + b*c)/d))/(d*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + d*tan(3/2
*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + d*tan(3/2*a)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + d*tan(3/2*a)^2*tan(3/2*b*
c/d)^2*tan(1/2*b*c/d)^2 + d*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + d*tan(3/2*a)^2*tan(1/2*a)^2 + d*t
an(3/2*a)^2*tan(3/2*b*c/d)^2 + d*tan(1/2*a)^2*tan(3/2*b*c/d)^2 + d*tan(3/2*a)^2*tan(1/2*b*c/d)^2 + d*tan(1/2*a
)^2*tan(1/2*b*c/d)^2 + d*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + d*tan(3/2*a)^2 + d*tan(1/2*a)^2 + d*tan(3/2*b*c/d
)^2 + d*tan(1/2*b*c/d)^2 + d)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{c+d\,x} \,d x \]

[In]

int((cos(a + b*x)^2*sin(a + b*x))/(c + d*x),x)

[Out]

int((cos(a + b*x)^2*sin(a + b*x))/(c + d*x), x)