\(\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [267]

Optimal result

Integrand size = 28, antiderivative size = 128 \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right ) \sin \left (2 c-\frac {2 d e}{f}\right )}{2 a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \] Output:

cos(c-d*e/f)*Ci(d*e/f+d*x)/a/f-1/2*Ci(2*d*e/f+2*d*x)*sin(2*c-2*d*e/f)/a/f- 
sin(c-d*e/f)*Si(d*e/f+d*x)/a/f-1/2*cos(2*c-2*d*e/f)*Si(2*d*e/f+2*d*x)/a/f
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=-\frac {-2 \cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (d \left (\frac {e}{f}+x\right )\right )+\operatorname {CosIntegral}\left (\frac {2 d (e+f x)}{f}\right ) \sin \left (2 c-\frac {2 d e}{f}\right )+2 \sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f} \] Input:

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

Output:

-1/2*(-2*Cos[c - (d*e)/f]*CosIntegral[d*(e/f + x)] + CosIntegral[(2*d*(e + 
 f*x))/f]*Sin[2*c - (2*d*e)/f] + 2*Sin[c - (d*e)/f]*SinIntegral[d*(e/f + x 
)] + Cos[2*c - (2*d*e)/f]*SinIntegral[(2*d*(e + f*x))/f])/(a*f)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5034, 3042, 3784, 3042, 3780, 3783, 4906, 27, 3042, 3784, 3042, 3780, 3783}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{(e+f x) (a \sin (c+d x)+a)} \, dx\)

\(\Big \downarrow \) 5034

\(\displaystyle \frac {\int \frac {\cos (c+d x)}{e+f x}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{e+f x}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {\cos \left (c-\frac {d e}{f}\right ) \int \frac {\cos \left (\frac {d e}{f}+d x\right )}{e+f x}dx-\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx-\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{e+f x}dx}{a}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{2 (e+f x)}dx}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{e+f x}dx}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{e+f x}dx}{2 a}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\cos \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx+\cos \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x+\frac {\pi }{2}\right )}{e+f x}dx+\cos \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx}{2 a}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x+\frac {\pi }{2}\right )}{e+f x}dx+\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{f}}{2 a}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}}{a}-\frac {\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{f}+\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{f}}{2 a}\)

Input:

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

Output:

((Cos[c - (d*e)/f]*CosIntegral[(d*e)/f + d*x])/f - (Sin[c - (d*e)/f]*SinIn 
tegral[(d*e)/f + d*x])/f)/a - ((CosIntegral[(2*d*e)/f + 2*d*x]*Sin[2*c - ( 
2*d*e)/f])/f + (Cos[2*c - (2*d*e)/f]*SinIntegral[(2*d*e)/f + 2*d*x])/f)/(2 
*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

rule 5034

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.) 
*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Cos[c + 
d*x]^(n - 2), x], x] - Simp[1/b   Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[ 
c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^ 
2 - b^2, 0]
 

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(-\frac {\frac {-\operatorname {Si}\left (-2 d x -2 c +\frac {2 c f -2 d e}{f}\right ) \cos \left (\frac {2 c f -2 d e}{f}\right )+\operatorname {Ci}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right ) \sin \left (\frac {2 c f -2 d e}{f}\right )}{2 f}+\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}-\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{a}\)	\(163\)
default	\(-\frac {\frac {-\operatorname {Si}\left (-2 d x -2 c +\frac {2 c f -2 d e}{f}\right ) \cos \left (\frac {2 c f -2 d e}{f}\right )+\operatorname {Ci}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right ) \sin \left (\frac {2 c f -2 d e}{f}\right )}{2 f}+\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}-\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{a}\)	\(163\)
risch	\(-\frac {{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a f}-\frac {{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{\frac {2 i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (-2 i d x -2 i c -\frac {2 \left (-i c f +i d e \right )}{f}\right )}{4 a f}+\frac {i {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (2 i d x +2 i c -\frac {2 i \left (c f -d e \right )}{f}\right )}{4 a f}\)	\(204\)

Input:

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

Output:

-1/a*(1/2/f*(-Si(-2*d*x-2*c+2*(c*f-d*e)/f)*cos(2*(c*f-d*e)/f)+Ci(2*d*x+2*c 
-2*(c*f-d*e)/f)*sin(2*(c*f-d*e)/f))+Si(-d*x-c-(-c*f+d*e)/f)*sin((-c*f+d*e) 
/f)/f-Ci(d*x+c+(-c*f+d*e)/f)*cos((-c*f+d*e)/f)/f)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {2 \, \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) - \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) - \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) - 2 \, \sin \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right )}{2 \, a f} \] Input:

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

Output:

1/2*(2*cos(-(d*e - c*f)/f)*cos_integral((d*f*x + d*e)/f) - cos_integral(2* 
(d*f*x + d*e)/f)*sin(-2*(d*e - c*f)/f) - cos(-2*(d*e - c*f)/f)*sin_integra 
l(2*(d*f*x + d*e)/f) - 2*sin(-(d*e - c*f)/f)*sin_integral((d*f*x + d*e)/f) 
)/(a*f)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.20 \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=-\frac {2 \, d {\left (E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - d {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) + 2 \, d {\left (-i \, E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + i \, E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - d {\left (E_{1}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}{4 \, a d f} \] Input:

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

Output:

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

Giac [C] (verification not implemented)

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

Time = 0.40 (sec) , antiderivative size = 4510, normalized size of antiderivative = 35.23 \[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(3*pi + 3*pi*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 2*imag_part 
(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 
 + 2*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2*t 
an(1/2*d*e/f)^2 - 4*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^4*tan( 
d*e/f)^2*tan(1/2*d*e/f)^2 - 4*real_part(cos_integral(-d*x - d*e/f))*tan(1/ 
2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 4*sin_integral(2*(d*f*x + d*e)/f)*t 
an(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 8*imag_part(cos_integral(d*x + 
 d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) - 8*imag_part(cos_integr 
al(-d*x - d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) + 16*sin_integr 
al((d*f*x + d*e)/f)*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) - 4*real_part 
(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^4*tan(d*e/f)*tan(1/2*d*e/f)^2 - 
 4*real_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^4*tan(d*e/f)*tan(1 
/2*d*e/f)^2 - 8*imag_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^3*tan(d*e/ 
f)^2*tan(1/2*d*e/f)^2 + 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c) 
^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 8*real_part(cos_integral(2*d*x + 2*d*e/ 
f))*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 8*real_part(cos_integral( 
-2*d*x - 2*d*e/f))*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 16*sin_int 
egral((d*f*x + d*e)/f)*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 3*pi*t 
an(1/2*c)^4*tan(d*e/f)^2 - 2*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan( 
1/2*c)^4*tan(d*e/f)^2 + 2*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\cos \left (d x +c \right )}{f x +e}d x -\left (\int \frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{f x +e}d x \right )}{a} \] Input:

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

Output:

(int(cos(c + d*x)/(e + f*x),x) - int((cos(c + d*x)*sin(c + d*x))/(e + f*x) 
,x))/a