\(\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx\) [141]
Optimal result
Integrand size = 22, antiderivative size = 129 \[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{8 d}+\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{4 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}+\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}
\]
[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] (verified)
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4491, 3384, 3380, 3383}
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{4 d}+\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}
\]
[In]
Int[(Cos[a + b*x]^3*Sin[a + b*x])/(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 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 (2 a+2 b x)}{4 (c+d x)}+\frac {\sin (4 a+4 b x)}{8 (c+d x)}\right ) \, dx \\ & = \frac {1}{8} \int \frac {\sin (4 a+4 b x)}{c+d x} \, dx+\frac {1}{4} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx \\ & = \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+\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 {\operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{8 d}+\frac {\operatorname {CosIntegral}\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] (verified)
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )+2 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{8 d}
\]
[In]
Integrate[(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x),x]
[Out]
(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])/
(8*d)
Maple [A] (verified)
Time = 0.97 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {\frac {b \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{8}+\frac {b \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{32}}{b}\) |
\(178\) |
| default |
\(\frac {\frac {b \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{8}+\frac {b \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{32}}{b}\) |
\(178\) |
| risch |
\(-\frac {i {\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{16 d}-\frac {i {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{8 d}+\frac {i {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{8 d}+\frac {i {\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{16 d}\) |
\(194\) |
| | |
|
|
|
|
[In]
int(cos(b*x+a)^3*sin(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
[Out]
1/b*(1/8*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)+1/32*b*(-4*Si(-4*x*b-4*a-4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d-4*Ci(4*x*b+4*a+4*(-a*d+b*c)/d)*sin(4
*(-a*d+b*c)/d)/d))
Fricas [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {2 \, \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + 2 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{8 \, d}
\]
[In]
integrate(cos(b*x+a)^3*sin(b*x+a)/(d*x+c),x, algorithm="fricas")
[Out]
1/8*(2*cos_integral(2*(b*d*x + b*c)/d)*sin(-2*(b*c - a*d)/d) + cos_integral(4*(b*d*x + b*c)/d)*sin(-4*(b*c - a
*d)/d) + cos(-4*(b*c - a*d)/d)*sin_integral(4*(b*d*x + b*c)/d) + 2*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x
+ b*c)/d))/d
Sympy [F]
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{c + d x}\, dx
\]
[In]
integrate(cos(b*x+a)**3*sin(b*x+a)/(d*x+c),x)
[Out]
Integral(sin(a + b*x)*cos(a + b*x)**3/(c + d*x), x)
Maxima [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.18
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {2 \, b {\left (i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (-i \, E_{1}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (E_{1}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )}{16 \, b d}
\]
[In]
integrate(cos(b*x+a)^3*sin(b*x+a)/(d*x+c),x, algorithm="maxima")
[Out]
1/16*(2*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, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I
*exp_integral_e(1, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-4*(b*c - a*d)/d) - 2*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, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -4*(-I*b*c - I*(b*x + a
)*d + I*a*d)/d))*sin(-4*(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 = 0.54 (sec) , antiderivative size = 6046, normalized size of antiderivative = 46.87
\[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(b*x+a)^3*sin(b*x+a)/(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(c
os_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*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) + 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))*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 + 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*ta
n(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_integral
(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))*t
an(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_in
tegral(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*si
n_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_in
tegral(-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 - ima
g_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_i
ntegral(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*t
an(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*r
eal_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_int
egral(-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_part
(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))*t
an(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_par
t(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(co
s_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))*ta
n(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*sin
_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*i
mag_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) + ima
g_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))*t
an(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(c
os_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_part
(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_pa
rt(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_par
t(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(co
s_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_int
egral(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_integr
al(-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*rea
l_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_integr
al(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_part
(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))*ta
n(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_i
ntegral(-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*ima
g_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*s
in_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_integ
ral(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_integra
l(-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))*ta
n(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(co
s_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 + ima
g_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*s
in_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)*tan
(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))*t
an(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))*t
an(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_par
t(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*tan
(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*t
an(a)^2 + d*tan(2*b*c/d)^2 + d*tan(b*c/d)^2 + d)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3\,\sin \left (a+b\,x\right )}{c+d\,x} \,d x
\]
[In]
int((cos(a + b*x)^3*sin(a + b*x))/(c + d*x),x)
[Out]
int((cos(a + b*x)^3*sin(a + b*x))/(c + d*x), x)