\(\int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [300]

Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a (e+f x)^2}{2 b^2 f}+\frac {(e+f x) \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {\sqrt {a^2-b^2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {\sqrt {a^2-b^2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {f \sin (c+d x)}{b d^2} \] Output:

1/2*a*(f*x+e)^2/b^2/f+(f*x+e)*cos(d*x+c)/b/d+I*(a^2-b^2)^(1/2)*(f*x+e)*ln( 
1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^2/d-I*(a^2-b^2)^(1/2)*(f*x+e)* 
ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^2/d+(a^2-b^2)^(1/2)*f*polyl 
og(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^2/d^2-(a^2-b^2)^(1/2)*f*pol 
ylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^2/d^2-f*sin(d*x+c)/b/d^2
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1991\) vs. \(2(297)=594\).

Time = 16.39 (sec) , antiderivative size = 1991, normalized size of antiderivative = 6.70 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(a*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x)))/(2*b^2*d^2) + ((d*e - c*f + f* 
(c + d*x))*Cos[c + d*x])/(b*d^2) - (f*Sin[c + d*x])/(b*d^2) + (((2*(d*e - 
c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - ( 
I*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x 
)/2])/(I*a + b - Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] + (I*f*Log[1 - I*Tan 
[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b 
+ Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] - (I*f*Log[1 - I*Tan[(c + d*x)/2]] 
*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + 
 b^2])])/Sqrt[-a^2 + b^2] + (I*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt 
[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])])/Sqrt[-a^ 
2 + b^2] - (I*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[- 
a^2 + b^2]))])/Sqrt[-a^2 + b^2] + (I*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/ 
2]))/(a - I*(b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] + (I*f*PolyLog[2, ( 
a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] 
- (I*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2] 
))])/Sqrt[-a^2 + b^2])*(((-a^2 + b^2)*e)/(b^2*(a + b*Sin[c + d*x])) - ((-a 
^2 + b^2)*c*f)/(b^2*d*(a + b*Sin[c + d*x])) + ((-a^2 + b^2)*f*(c + d*x))/( 
b^2*d*(a + b*Sin[c + d*x]))))/(d*((f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/ 
(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[-a^2 + b^2]*(1 
 - I*Tan[(c + d*x)/2])) + (f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d...
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5036, 17, 3042, 3777, 3042, 3117, 3804, 2694, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5036

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {2 \left (a^2-b^2\right ) \left (\frac {i b \left (\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {f \int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {f \int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}+\frac {a (e+f x)^2}{2 b^2 f}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{b}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {2 \left (a^2-b^2\right ) \left (\frac {i b \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{b d^2}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{b d^2}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}+\frac {a (e+f x)^2}{2 b^2 f}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2 \left (a^2-b^2\right ) \left (\frac {i b \left (\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}+\frac {a (e+f x)^2}{2 b^2 f}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{b}\)

Input:

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

Output:

(a*(e + f*x)^2)/(2*b^2*f) - (2*(a^2 - b^2)*(((-1/2*I)*b*(((e + f*x)*Log[1 
- (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) - (I*f*PolyLog[2, (I 
*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^2)))/Sqrt[a^2 - b^2] + (( 
I/2)*b*(((e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/( 
b*d) - (I*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^ 
2)))/Sqrt[a^2 - b^2]))/b^2 - (-(((e + f*x)*Cos[c + d*x])/d) + (f*Sin[c + d 
*x])/d^2)/b
 

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 27

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 5036

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (265 ) = 530\).

Time = 2.25 (sec) , antiderivative size = 1105, normalized size of antiderivative = 3.72

method	result	size
risch	\(\text {Expression too large to display}\)	\(1105\)

Input:

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

Output:

1/2*a/b^2*f*x^2+a/b^2*e*x+1/2*(d*x*f+I*f+d*e)/d^2/b*exp(I*(d*x+c))+1/2*(d* 
x*f-I*f+d*e)/d^2/b*exp(-I*(d*x+c))+2*I/d*e/(-a^2+b^2)^(1/2)*arctan(1/2*(2* 
I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))-2*I/d/b^2*a^2*e/(-a^2+b^2)^(1/2) 
*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))+1/d*f/(-a^2+b^2)^ 
(1/2)*ln((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2))) 
*x-1/d^2*f/(-a^2+b^2)^(1/2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I* 
a+(-a^2+b^2)^(1/2)))*c+1/d/b^2*a^2*f/(-a^2+b^2)^(1/2)*ln((I*a+exp(I*(d*x+c 
))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x+1/d^2*f/(-a^2+b^2)^(1/2)* 
ln((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*c-I/d 
^2*f/(-a^2+b^2)^(1/2)*dilog((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(-I*a 
+(-a^2+b^2)^(1/2)))-I/d^2/b^2*a^2*f/(-a^2+b^2)^(1/2)*dilog((I*a+exp(I*(d*x 
+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+2*I/d^2/b^2*f*a^2*c/(-a^2 
+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))+I/d^2/ 
b^2*a^2*f/(-a^2+b^2)^(1/2)*dilog((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/ 
(-I*a+(-a^2+b^2)^(1/2)))-1/d^2/b^2*a^2*f/(-a^2+b^2)^(1/2)*ln((-I*a-exp(I*( 
d*x+c))*b+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*c+1/d^2/b^2*a^2*f/(-a 
^2+b^2)^(1/2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^( 
1/2)))*c-1/d/b^2*a^2*f/(-a^2+b^2)^(1/2)*ln((-I*a-exp(I*(d*x+c))*b+(-a^2+b^ 
2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*x-1/d*f/(-a^2+b^2)^(1/2)*ln((I*a+exp(I* 
(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x-2*I/d^2*f*c/(-a^...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (257) = 514\).

Time = 0.22 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.48 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(a*d^2*f*x^2 + 2*a*d^2*e*x - I*b*f*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*c 
os(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-( 
a^2 - b^2)/b^2) - b)/b + 1) + I*b*f*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos( 
d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 
 - b^2)/b^2) - b)/b + 1) + I*b*f*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d* 
x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - 
 b^2)/b^2) - b)/b + 1) - I*b*f*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x 
+ c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b 
^2)/b^2) - b)/b + 1) - 2*b*f*sin(d*x + c) - (b*d*e - b*c*f)*sqrt(-(a^2 - b 
^2)/b^2)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2) 
/b^2) + 2*I*a) - (b*d*e - b*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + 
c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + (b*d*e - b 
*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 
2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) + (b*d*e - b*c*f)*sqrt(-(a^2 - b^2)/b^ 
2)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) 
 - 2*I*a) + (b*d*f*x + b*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*cos(d*x + c 
) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2) 
/b^2) - b)/b) - (b*d*f*x + b*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*cos(d*x 
 + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - 
b^2)/b^2) - b)/b) + (b*d*f*x + b*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} d e +2 \cos \left (d x +c \right ) a^{3} b d e -4 \cos \left (d x +c \right ) a^{2} b^{2} f +2 \cos \left (d x +c \right ) a \,b^{3} d f x +4 \cos \left (d x +c \right ) b^{4} f +16 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) x}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a}d x \right ) a^{2} b^{3} d^{2} f -16 \left (\int \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) x}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a}d x \right ) b^{5} d^{2} f +8 \left (\int \frac {x}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a}d x \right ) a^{3} b^{2} d^{2} f -8 \left (\int \frac {x}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a}d x \right ) a \,b^{4} d^{2} f -4 \sin \left (d x +c \right ) a^{2} b^{2} d f x -2 \sin \left (d x +c \right ) a \,b^{3} f +4 \sin \left (d x +c \right ) b^{4} d f x +2 a^{4} d^{2} e x -2 a^{3} b d e -a^{2} b^{2} d^{2} f \,x^{2}+4 a^{2} b^{2} f +2 b^{4} d^{2} f \,x^{2}-4 b^{4} f}{2 a^{3} b^{2} d^{2}} \] Input:

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

Output:

( - 4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a 
**3*d*e + 2*cos(c + d*x)*a**3*b*d*e - 4*cos(c + d*x)*a**2*b**2*f + 2*cos(c 
 + d*x)*a*b**3*d*f*x + 4*cos(c + d*x)*b**4*f + 16*int((tan((c + d*x)/2)*x) 
/(tan((c + d*x)/2)**4*a + 2*tan((c + d*x)/2)**3*b + 2*tan((c + d*x)/2)**2* 
a + 2*tan((c + d*x)/2)*b + a),x)*a**2*b**3*d**2*f - 16*int((tan((c + d*x)/ 
2)*x)/(tan((c + d*x)/2)**4*a + 2*tan((c + d*x)/2)**3*b + 2*tan((c + d*x)/2 
)**2*a + 2*tan((c + d*x)/2)*b + a),x)*b**5*d**2*f + 8*int(x/(tan((c + d*x) 
/2)**4*a + 2*tan((c + d*x)/2)**3*b + 2*tan((c + d*x)/2)**2*a + 2*tan((c + 
d*x)/2)*b + a),x)*a**3*b**2*d**2*f - 8*int(x/(tan((c + d*x)/2)**4*a + 2*ta 
n((c + d*x)/2)**3*b + 2*tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a), 
x)*a*b**4*d**2*f - 4*sin(c + d*x)*a**2*b**2*d*f*x - 2*sin(c + d*x)*a*b**3* 
f + 4*sin(c + d*x)*b**4*d*f*x + 2*a**4*d**2*e*x - 2*a**3*b*d*e - a**2*b**2 
*d**2*f*x**2 + 4*a**2*b**2*f + 2*b**4*d**2*f*x**2 - 4*b**4*f)/(2*a**3*b**2 
*d**2)