\(\int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx\) [159]

Optimal result

Integrand size = 24, antiderivative size = 129 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{32 d}+\frac {3 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{32 d}+\frac {3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{32 d}-\frac {\cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{32 d} \]

[Out]

3/32*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d-1/32*cos(6*a-6*b*c/d)*Si(6*b*c/d+6*b*x)/d-1/32*Ci(6*b*c/d+6*b*x)*sin
(6*a-6*b*c/d)/d+3/32*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4491, 3384, 3380, 3383} \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {\sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right )}{32 d}+\frac {3 \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{32 d}+\frac {3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{32 d}-\frac {\cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{32 d} \]

[In]

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

[Out]

-1/32*(CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d])/d + (3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2
*b*c)/d])/(32*d) + (3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(32*d) - (Cos[6*a - (6*b*c)/d]*SinI
ntegral[(6*b*c)/d + 6*b*x])/(32*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 {3 \sin (2 a+2 b x)}{32 (c+d x)}-\frac {\sin (6 a+6 b x)}{32 (c+d x)}\right ) \, dx \\ & = -\left (\frac {1}{32} \int \frac {\sin (6 a+6 b x)}{c+d x} \, dx\right )+\frac {3}{32} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx \\ & = -\left (\frac {1}{32} \cos \left (6 a-\frac {6 b c}{d}\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{c+d x} \, dx\right )+\frac {1}{32} \left (3 \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-\frac {1}{32} \sin \left (6 a-\frac {6 b c}{d}\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{c+d x} \, dx+\frac {1}{32} \left (3 \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 \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{32 d}+\frac {3 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{32 d}+\frac {3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{32 d}-\frac {\cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{32 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {6 b (c+d x)}{d}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )-3 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )-3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+\cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b (c+d x)}{d}\right )}{32 d} \]

[In]

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

[Out]

-1/32*(CosIntegral[(6*b*(c + d*x))/d]*Sin[6*a - (6*b*c)/d] - 3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c
)/d] - 3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x)
)/d])/d

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {\frac {3 b \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{64}-\frac {b \left (-\frac {6 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}-\frac {6 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}\right )}{192}}{b}\)	\(178\)
default	\(\frac {\frac {3 b \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{64}-\frac {b \left (-\frac {6 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}-\frac {6 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}\right )}{192}}{b}\)	\(178\)
risch	\(\frac {i {\mathrm e}^{-\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (6 i b x +6 i a -\frac {6 i \left (a d -c b \right )}{d}\right )}{64 d}-\frac {3 i {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{64 d}+\frac {3 i {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{64 d}-\frac {i {\mathrm e}^{\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-6 i b x -6 i a -\frac {6 \left (-i a d +i c b \right )}{d}\right )}{64 d}\)	\(194\)

[In]

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

[Out]

1/b*(3/64*b*(-2*Si(-2*x*b-2*a-2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d-2*Ci(2*x*b+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d
+b*c)/d)/d)-1/192*b*(-6*Si(-6*x*b-6*a-6*(-a*d+b*c)/d)*cos(6*(-a*d+b*c)/d)/d-6*Ci(6*x*b+6*a+6*(-a*d+b*c)/d)*sin
(6*(-a*d+b*c)/d)/d))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\frac {3 \, \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \operatorname {Ci}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) + 3 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{32 \, d} \]

[In]

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

[Out]

1/32*(3*cos_integral(2*(b*d*x + b*c)/d)*sin(-2*(b*c - a*d)/d) - cos_integral(6*(b*d*x + b*c)/d)*sin(-6*(b*c -
a*d)/d) - cos(-6*(b*c - a*d)/d)*sin_integral(6*(b*d*x + b*c)/d) + 3*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*
x + b*c)/d))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.18 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {3 \, b {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (-i \, E_{1}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (E_{1}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )}{64 \, b d} \]

[In]

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

[Out]

-1/64*(3*b*(-I*exp_integral_e(1, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(1, -2*(-I*b*c - I*(b
*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) - b*(-I*exp_integral_e(1, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) +
I*exp_integral_e(1, -6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-6*(b*c - a*d)/d) + 3*b*(exp_integral_e(1, 2*
(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c -
a*d)/d) - b*(exp_integral_e(1, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -6*(-I*b*c - I*(b*x +
a)*d + I*a*d)/d))*sin(-6*(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 = 0.58 (sec) , antiderivative size = 6046, normalized size of antiderivative = 46.87 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/64*(imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(
cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b
*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan
(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c
/d)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*rea
l_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral
(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))
*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)
^2*tan(3*b*c/d)*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan
(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real
_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(
-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(6*b*x + 6*b*c/d))*ta
n(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)
^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 - imag_part(cos_integral(-
6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*t
an(3*b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2 - 12*imag_part(cos_integr
al(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) + 12*imag_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)*tan(3*b*c
/d)^2*tan(b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 3*imag_part(cos
_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan
(3*a)^2*tan(a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 2
*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)
^2*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d
)^2 - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 8*sin_integral
(6*(b*d*x + b*c)/d)*tan(3*a)*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + imag_part(cos_integral(6*b*x + 6*b*c/d))*tan
(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan
(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos
_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*
a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 -
imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3
*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*si
n_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^
2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d) +
2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2*tan(3*b*c/d) - 6*real_part(cos_integral(2*b*x
+ 2*b*c/d))*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2 - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*t
an(3*b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2 - 2*real_part(cos_
integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(
3*a)^2*tan(a)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2*tan(b*c/d) + 6*re
al_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x
- 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(3*b
*c/d)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_p
art(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(3*a)^2*tan(a)*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(b*c/d)^2 +
2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x +
6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)*tan(b*c/d)^2 + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3
*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_
part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x + 6*b
*c/d))*tan(3*a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d
)^2*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*real_part
(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b*c/d))
*tan(3*a)^2*tan(a)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)^2 - 3*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(a)^2 - 2*
sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(a)^2 +
4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d) - 4*imag_part(cos_integral(-6*b*x -
6*b*c/d))*tan(3*a)*tan(a)^2*tan(3*b*c/d) + 8*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)*tan(a)^2*tan(3*b*c/d) +
imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d)
)*tan(3*a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 - imag_par
t(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2*tan
(3*b*c/d)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(3*b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b
*c/d))*tan(a)^2*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2 - 3*imag_p
art(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(3*b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)
^2*tan(3*b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(3*b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)
/d)*tan(a)^2*tan(3*b*c/d)^2 - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d) + 12*im
ag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a)*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*tan(
3*a)^2*tan(a)*tan(b*c/d) - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) + 12*i
mag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)
*tan(a)*tan(3*b*c/d)^2*tan(b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 + 3*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(
3*a)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(b*c/d)^2 - 2*sin_integral(6*(b*
d*x + b*c)/d)*tan(3*a)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2*tan(b*c/d)^2 + imag_part(
cos_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(b*c/d)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2*tan
(b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*x
- 6*b*c/d))*tan(a)^2*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 - 6*sin_integral(2
*(b*d*x + b*c)/d)*tan(a)^2*tan(b*c/d)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(3*b*c/d)*tan
(b*c/d)^2 - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d)*tan(b*c/d)^2 + 8*sin_integral(6*
(b*d*x + b*c)/d)*tan(3*a)*tan(3*b*c/d)*tan(b*c/d)^2 - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d)^2*
tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*imag_part(cos_integr
al(-2*b*x - 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)^2*t
an(b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d
)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2*tan(a) - 6*real_part(cos
_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(a) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(a)^2
+ 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(a)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*t
an(3*a)^2*tan(3*b*c/d) + 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^2*tan(3*b*c/d) - 2*real_part(cos
_integral(6*b*x + 6*b*c/d))*tan(a)^2*tan(3*b*c/d) - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2*tan(3
*b*c/d) - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(3*b*c/d)^2 - 2*real_part(cos_integral(-6*b*x
- 6*b*c/d))*tan(3*a)*tan(3*b*c/d)^2 - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(3*b*c/d)^2 - 6*re
al_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(3*b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan
(3*a)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2*tan(b*c/d) - 6*real_part(cos_integ
ral(2*b*x + 2*b*c/d))*tan(a)^2*tan(b*c/d) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)^2*tan(b*c/d) +
6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*b*c
/d))*tan(3*b*c/d)^2*tan(b*c/d) + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)*tan(b*c/d)^2 + 2*real_par
t(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(b*c/d)^2 + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*ta
n(b*c/d)^2 + 6*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d)^2 - 2*real_part(cos_integral(6*b*x
+ 6*b*c/d))*tan(3*b*c/d)*tan(b*c/d)^2 - 2*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)*tan(b*c/d)^2
- imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a)^2 - 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(3*a)^2
+ 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*a)^2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)^
2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*a)^2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*a)^2 + imag_part(co
s_integral(6*b*x + 6*b*c/d))*tan(a)^2 + 3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)^2 - 3*imag_part(cos_
integral(-2*b*x - 2*b*c/d))*tan(a)^2 - imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(a)^2 + 2*sin_integral(6*(
b*d*x + b*c)/d)*tan(a)^2 + 6*sin_integral(2*(b*d*x + b*c)/d)*tan(a)^2 + 4*imag_part(cos_integral(6*b*x + 6*b*c
/d))*tan(3*a)*tan(3*b*c/d) - 4*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*a)*tan(3*b*c/d) + 8*sin_integra
l(6*(b*d*x + b*c)/d)*tan(3*a)*tan(3*b*c/d) - imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d)^2 - 3*imag_
part(cos_integral(2*b*x + 2*b*c/d))*tan(3*b*c/d)^2 + 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*b*c/d)^
2 + imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*c/d)^2 - 2*sin_integral(6*(b*d*x + b*c)/d)*tan(3*b*c/d)^
2 - 6*sin_integral(2*(b*d*x + b*c)/d)*tan(3*b*c/d)^2 - 12*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(a)*tan(
b*c/d) + 12*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(a)*tan(b*c/d) - 24*sin_integral(2*(b*d*x + b*c)/d)*t
an(a)*tan(b*c/d) + imag_part(cos_integral(6*b*x + 6*b*c/d))*tan(b*c/d)^2 + 3*imag_part(cos_integral(2*b*x + 2*
b*c/d))*tan(b*c/d)^2 - 3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*c/d)^2 - imag_part(cos_integral(-6*b*
x - 6*b*c/d))*tan(b*c/d)^2 + 2*sin_integral(6*(b*d*x + b*c)/d)*tan(b*c/d)^2 + 6*sin_integral(2*(b*d*x + b*c)/d
)*tan(b*c/d)^2 + 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*a) + 2*real_part(cos_integral(-6*b*x - 6*b*c
/d))*tan(3*a) - 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(a) - 6*real_part(cos_integral(-2*b*x - 2*b*c/d)
)*tan(a) - 2*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*c/d) - 2*real_part(cos_integral(-6*b*x - 6*b*c/d
))*tan(3*b*c/d) + 6*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*c/d) + 6*real_part(cos_integral(-2*b*x - 2*
b*c/d))*tan(b*c/d) + imag_part(cos_integral(6*b*x + 6*b*c/d)) - 3*imag_part(cos_integral(2*b*x + 2*b*c/d)) + 3
*imag_part(cos_integral(-2*b*x - 2*b*c/d)) - imag_part(cos_integral(-6*b*x - 6*b*c/d)) + 2*sin_integral(6*(b*d
*x + b*c)/d) - 6*sin_integral(2*(b*d*x + b*c)/d))/(d*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3
*a)^2*tan(a)^2*tan(3*b*c/d)^2 + d*tan(3*a)^2*tan(a)^2*tan(b*c/d)^2 + d*tan(3*a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2
+ d*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3*a)^2*tan(a)^2 + d*tan(3*a)^2*tan(3*b*c/d)^2 + d*tan(a)^2*ta
n(3*b*c/d)^2 + d*tan(3*a)^2*tan(b*c/d)^2 + d*tan(a)^2*tan(b*c/d)^2 + d*tan(3*b*c/d)^2*tan(b*c/d)^2 + d*tan(3*a
)^2 + d*tan(a)^2 + d*tan(3*b*c/d)^2 + d*tan(b*c/d)^2 + d)

Mupad [F(-1)]

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

[In]

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

[Out]

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