\(\int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [262]

Optimal result

Integrand size = 28, antiderivative size = 95 \[ \int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {1}{a f (e+f x)}-\frac {d \cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {\sin (c+d x)}{a f (e+f x)}+\frac {d \sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {-d (e+f x) \cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (d \left (\frac {e}{f}+x\right )\right )+f (-1+\sin (c+d x))+d (e+f x) \sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )}{a f^2 (e+f x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {5034, 17, 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.

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

\(\Big \downarrow \) 5034

\(\displaystyle \frac {\int \frac {1}{(e+f x)^2}dx}{a}-\frac {\int \frac {\sin (c+d x)}{(e+f x)^2}dx}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {\int \frac {\sin (c+d x)}{(e+f x)^2}dx}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin (c+d x)}{(e+f x)^2}dx}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {\frac {d \int \frac {\cos (c+d x)}{e+f x}dx}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {d \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{e+f x}dx}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {\frac {d \left (\cos \left (c-\frac {d e}{f}\right ) \int \frac {\cos \left (\frac {d e}{f}+d x\right )}{e+f x}dx-\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {d \left (\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx-\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {\frac {d \left (\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {\frac {d \left (\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\sin (c+d x)}{f (e+f x)}}{a}-\frac {1}{a f (e+f x)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

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 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 = 1.01 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(\frac {d \left (\frac {\sin \left (d x +c \right )}{\left (-c f +d e +f \left (d x +c \right )\right ) f}-\frac {-\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}+\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{f}-\frac {1}{\left (-c f +d e +f \left (d x +c \right )\right ) f}\right )}{a}\)	\(137\)
default	\(\frac {d \left (\frac {\sin \left (d x +c \right )}{\left (-c f +d e +f \left (d x +c \right )\right ) f}-\frac {-\frac {\operatorname {Si}\left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}+\frac {\operatorname {Ci}\left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{f}-\frac {1}{\left (-c f +d e +f \left (d x +c \right )\right ) f}\right )}{a}\)	\(137\)
risch	\(-\frac {1}{a f \left (f x +e \right )}+\frac {d \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (-i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a \,f^{2}}+\frac {d \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \operatorname {expIntegral}_{1}\left (i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}+\frac {\left (-2 d x f -2 d e \right ) \sin \left (d x +c \right )}{2 f a \left (f x +e \right ) \left (-d x f -d e \right )}\)	\(162\)

Input:

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

Output:

d/a*(sin(d*x+c)/(-c*f+d*e+f*(d*x+c))/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)/f-1/(-c*f+d*e+f*(d* 
x+c))/f)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {{\left (d f x + d e\right )} \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) - {\left (d f x + d e\right )} \sin \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right ) - f \sin \left (d x + c\right ) + f}{a f^{3} x + a e f^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {d^{2} {\left (i \, E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) - i \, E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) + d^{2} {\left (E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - 2 \, d^{2}}{2 \, {\left (a d e f + {\left (d x + c\right )} a f^{2} - a c f^{2}\right )} d} \] Input:

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

Output:

1/2*(d^2*(I*exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) - I*exp_i 
ntegral_e(2, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*cos(-(d*e - c*f)/f) + d^ 
2*(exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + exp_integral_e(2 
, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*sin(-(d*e - c*f)/f) - 2*d^2)/((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 = 0.58 (sec) , antiderivative size = 3192, normalized size of antiderivative = 33.60 \[ \int \frac {\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(d*f*x*real_part(cos_integral(d*x + d*e/f))*tan(1/2*d*x)^2*tan(1/2*c) 
^2*tan(1/2*d*e/f)^2 + d*f*x*real_part(cos_integral(-d*x - d*e/f))*tan(1/2* 
d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f)^2 - 2*d*f*x*imag_part(cos_integral(d*x 
+ d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f) + 2*d*f*x*imag_part(c 
os_integral(-d*x - d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f) - 4* 
d*f*x*sin_integral((d*f*x + d*e)/f)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d* 
e/f) + 2*d*f*x*imag_part(cos_integral(d*x + d*e/f))*tan(1/2*d*x)^2*tan(1/2 
*c)*tan(1/2*d*e/f)^2 - 2*d*f*x*imag_part(cos_integral(-d*x - d*e/f))*tan(1 
/2*d*x)^2*tan(1/2*c)*tan(1/2*d*e/f)^2 + 4*d*f*x*sin_integral((d*f*x + d*e) 
/f)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*d*e/f)^2 + d*e*real_part(cos_integra 
l(d*x + d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + d*e*real_pa 
rt(cos_integral(-d*x - d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f)^ 
2 - d*f*x*real_part(cos_integral(d*x + d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2 
 - d*f*x*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2 
 + 4*d*f*x*real_part(cos_integral(d*x + d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)* 
tan(1/2*d*e/f) + 4*d*f*x*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*d*x 
)^2*tan(1/2*c)*tan(1/2*d*e/f) - 2*d*e*imag_part(cos_integral(d*x + d*e/f)) 
*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f) + 2*d*e*imag_part(cos_integral 
(-d*x - d*e/f))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f) - 4*d*e*sin_int 
egral((d*f*x + d*e)/f)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*d*e/f) - d*f...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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