\(\int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [268]

Optimal result

Integrand size = 28, antiderivative size = 175 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {\cos (c+d x)}{a f (e+f x)}-\frac {d \cos \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}-\frac {d \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right ) \sin \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {\sin (2 c+2 d x)}{2 a f (e+f x)}-\frac {d \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {d \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {-2 f \cos (c+d x)-2 d (e+f x) \cos \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d (e+f x)}{f}\right )-2 d (e+f x) \operatorname {CosIntegral}\left (d \left (\frac {e}{f}+x\right )\right ) \sin \left (c-\frac {d e}{f}\right )+f \sin (2 (c+d x))-2 d e \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )-2 d f x \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+2 d e \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )+2 d f x \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f^2 (e+f x)} \] Input:

Integrate[Cos[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5034, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783, 4906, 27, 3042, 3778, 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.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)} \, dx\)

\(\Big \downarrow \) 5034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

Int[Cos[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

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 = 1.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {d \left (-\frac {\sin \left (2 d x +2 c \right )}{2 f \left (c f -d e -f \left (d x +c \right )\right )}-\frac {\operatorname {Si}\left (-2 d x -2 c +\frac {2 c f -2 d e}{f}\right ) \sin \left (\frac {2 c f -2 d e}{f}\right )+\operatorname {Ci}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right ) \cos \left (\frac {2 c f -2 d e}{f}\right )}{f^{2}}-\frac {\cos \left (d x +c \right )}{\left (-c f +d e +f \left (d x +c \right )\right ) f}-\frac {-\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}-\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}}{f}\right )}{a}\)	\(229\)
default	\(\frac {d \left (-\frac {\sin \left (2 d x +2 c \right )}{2 f \left (c f -d e -f \left (d x +c \right )\right )}-\frac {\operatorname {Si}\left (-2 d x -2 c +\frac {2 c f -2 d e}{f}\right ) \sin \left (\frac {2 c f -2 d e}{f}\right )+\operatorname {Ci}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right ) \cos \left (\frac {2 c f -2 d e}{f}\right )}{f^{2}}-\frac {\cos \left (d x +c \right )}{\left (-c f +d e +f \left (d x +c \right )\right ) f}-\frac {-\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}-\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}}{f}\right )}{a}\)	\(229\)
risch	\(\frac {i d \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}-\frac {i d \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a \,f^{2}}+\frac {d \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (-2 i d x -2 i c -\frac {2 \left (-i c f +i d e \right )}{f}\right )}{2 a \,f^{2}}+\frac {d \,{\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (2 i d x +2 i c -\frac {2 i \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}-\frac {\left (-2 d x f -2 d e \right ) \cos \left (d x +c \right )}{2 f a \left (f x +e \right ) \left (-d x f -d e \right )}+\frac {\left (-2 d x f -2 d e \right ) \sin \left (2 d x +2 c \right )}{4 f a \left (f x +e \right ) \left (-d x f -d e \right )}\)	\(297\)

Input:

int(cos(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {{\left (d f x + d e\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) - f \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (d f x + d e\right )} \operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) \sin \left (-\frac {d e - c f}{f}\right ) - {\left (d f x + d e\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) + {\left (d f x + d e\right )} \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right ) + f \cos \left (d x + c\right )}{a f^{3} x + a e f^{2}} \] Input:

integrate(cos(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

Output:

-((d*f*x + d*e)*cos(-2*(d*e - c*f)/f)*cos_integral(2*(d*f*x + d*e)/f) - f* 
cos(d*x + c)*sin(d*x + c) + (d*f*x + d*e)*cos_integral((d*f*x + d*e)/f)*si 
n(-(d*e - c*f)/f) - (d*f*x + d*e)*sin(-2*(d*e - c*f)/f)*sin_integral(2*(d* 
f*x + d*e)/f) + (d*f*x + d*e)*cos(-(d*e - c*f)/f)*sin_integral((d*f*x + d* 
e)/f) + f*cos(d*x + c))/(a*f^3*x + a*e*f^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3/(f*x+e)**2/(a+a*sin(d*x+c)),x)
 

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.77 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {2 \, d^{2} {\left (E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - d^{2} {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) + 2 \, d^{2} {\left (-i \, E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + i \, E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - d^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}{4 \, {\left (a d e f + {\left (d x + c\right )} a f^{2} - a c f^{2}\right )} d} \] Input:

integrate(cos(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

Output:

-1/4*(2*d^2*(exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + exp_in 
tegral_e(2, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*cos(-(d*e - c*f)/f) - d^2 
*(-I*exp_integral_e(2, 2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f) + I*exp_integ 
ral_e(2, -2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f))*cos(-2*(d*e - c*f)/f) + 2 
*d^2*(-I*exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + I*exp_inte 
gral_e(2, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*sin(-(d*e - c*f)/f) - d^2*( 
exp_integral_e(2, 2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f) + exp_integral_e(2 
, -2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f))*sin(-2*(d*e - c*f)/f))/((a*d*e*f 
 + (d*x + c)*a*f^2 - a*c*f^2)*d)
 

Giac [C] (verification not implemented)

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

Time = 2.12 (sec) , antiderivative size = 46878, normalized size of antiderivative = 267.87 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

-1/2*(d*f*x*imag_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2 
*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - d*f*x*imag_part(cos_integral 
(-d*x - d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/ 
2*d*e/f)^2 - d*f*x*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(d*x)^2*tan 
(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - d*f*x*real_part(c 
os_integral(-2*d*x - 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan( 
d*e/f)^2*tan(1/2*d*e/f)^2 + 2*d*f*x*sin_integral((d*f*x + d*e)/f)*tan(d*x) 
^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 2*d*f*x*rea 
l_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*t 
an(d*e/f)^2*tan(1/2*d*e/f) + 2*d*f*x*real_part(cos_integral(-d*x - d*e/f)) 
*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) + 2*d* 
f*x*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan 
(1/2*c)^4*tan(d*e/f)*tan(1/2*d*e/f)^2 - 2*d*f*x*imag_part(cos_integral(-2* 
d*x - 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)*tan(1/2* 
d*e/f)^2 + 4*d*f*x*sin_integral(2*(d*f*x + d*e)/f)*tan(d*x)^2*tan(1/2*d*x) 
^2*tan(1/2*c)^4*tan(d*e/f)*tan(1/2*d*e/f)^2 - 4*d*f*x*imag_part(cos_integr 
al(2*d*x + 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(d*e/f)^2*t 
an(1/2*d*e/f)^2 + 4*d*f*x*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(d* 
x)^2*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 2*d*f*x*r 
eal_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \] Input:

int(cos(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))),x)
 

Output:

int(cos(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\cos \left (d x +c \right )^{3}}{\sin \left (d x +c \right ) e^{2}+2 \sin \left (d x +c \right ) e f x +\sin \left (d x +c \right ) f^{2} x^{2}+e^{2}+2 e f x +f^{2} x^{2}}d x}{a} \] Input:

int(cos(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)
 

Output:

int(cos(c + d*x)**3/(sin(c + d*x)*e**2 + 2*sin(c + d*x)*e*f*x + sin(c + d* 
x)*f**2*x**2 + e**2 + 2*e*f*x + f**2*x**2),x)/a