\(\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [267]
Optimal result
Integrand size = 28, antiderivative size = 128 \[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right ) \sin \left (2 c-\frac {2 d e}{f}\right )}{2 a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}
\]
[Out]
Ci(d*e/f+d*x)*cos(c-d*e/f)/a/f-1/2*cos(2*c-2*d*e/f)*Si(2*d*e/f+2*d*x)/a/f-1/2*Ci(2*d*e/f+2*d*x)*sin(2*c-2*d*e/
f)/a/f-Si(d*e/f+d*x)*sin(c-d*e/f)/a/f
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4619, 3384, 3380, 3383, 4491,
12} \[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}
\]
[In]
Int[Cos[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
(Cos[c - (d*e)/f]*CosIntegral[(d*e)/f + d*x])/(a*f) - (CosIntegral[(2*d*e)/f + 2*d*x]*Sin[2*c - (2*d*e)/f])/(2
*a*f) - (Sin[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/(a*f) - (Cos[2*c - (2*d*e)/f]*SinIntegral[(2*d*e)/f + 2*
d*x])/(2*a*f)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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]
Rule 4619
Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*S
in[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \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} \\ & = -\frac {\int \frac {\sin (2 c+2 d x)}{2 (e+f x)} \, dx}{a}+\frac {\cos \left (c-\frac {d e}{f}\right ) \int \frac {\cos \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}-\frac {\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 {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\int \frac {\sin (2 c+2 d x)}{e+f x} \, dx}{2 a} \\ & = \frac {\cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{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 ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x\right )}{e+f x} \, dx}{2 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}{2 a} \\ & = \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} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.40 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.82
\[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=-\frac {-2 \cos \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (d \left (\frac {e}{f}+x\right )\right )+\operatorname {CosIntegral}\left (\frac {2 d (e+f x)}{f}\right ) \sin \left (2 c-\frac {2 d e}{f}\right )+2 \sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f}
\]
[In]
Integrate[Cos[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]
[Out]
-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*S
in[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)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.59
| | |
| method | result | size |
| | |
| risch |
\(-\frac {{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a f}-\frac {{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (-i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{\frac {2 i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (-2 i d x -2 i c -\frac {2 \left (-i c f +i d e \right )}{f}\right )}{4 a f}+\frac {i {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (2 i d x +2 i c -\frac {2 i \left (c f -d e \right )}{f}\right )}{4 a f}\) |
\(204\) |
| | |
|
|
|
|
[In]
int(cos(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
-1/2/a/f*exp(-I*(c*f-d*e)/f)*Ei(1,I*d*x+I*c-I*(c*f-d*e)/f)-1/2/a/f*exp(I*(c*f-d*e)/f)*Ei(1,-I*d*x-I*c-(-I*c*f+
I*d*e)/f)-1/4*I/a/f*exp(2*I*(c*f-d*e)/f)*Ei(1,-2*I*d*x-2*I*c-2*(-I*c*f+I*d*e)/f)+1/4*I/a/f*exp(-2*I*(c*f-d*e)/
f)*Ei(1,2*I*d*x+2*I*c-2*I*(c*f-d*e)/f)
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99
\[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {2 \, \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) - \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) - \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) - 2 \, \sin \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right )}{2 \, a f}
\]
[In]
integrate(cos(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
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_integral(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. \[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Exception raised: HeuristicGCDFailed}
\]
[In]
integrate(cos(d*x+c)**3/(f*x+e)/(a+a*sin(d*x+c)),x)
[Out]
Exception raised: HeuristicGCDFailed >> no luck
Maxima [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.20
\[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=-\frac {2 \, d {\left (E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - d {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) + 2 \, d {\left (-i \, E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + i \, E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - d {\left (E_{1}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}{4 \, a d f}
\]
[In]
integrate(cos(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/4*(2*d*(exp_integral_e(1, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + exp_integral_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_integra
l_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_
integral_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 = 0.57 (sec) , antiderivative size = 4510, normalized size of antiderivative = 35.23
\[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-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
*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*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)*tan(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_integral(-d*x - d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) + 16*sin_
integral((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*im
ag_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_integral((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*tan(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(1/2*c)^4*tan(d*e/f)^2 + 4*real_part(cos_integral(d*x + d*e/f))
*tan(1/2*c)^4*tan(d*e/f)^2 + 4*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^4*tan(d*e/f)^2 - 4*sin_integra
l(2*(d*f*x + d*e)/f)*tan(1/2*c)^4*tan(d*e/f)^2 - 16*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^3*tan(d*e/
f)^2*tan(1/2*d*e/f) - 16*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f) + 3*pi
*tan(1/2*c)^4*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(1/2*d*e/f)^2 - 2*
imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^4*tan(1/2*d*e/f)^2 - 4*real_part(cos_integral(d*x + d*e/f
))*tan(1/2*c)^4*tan(1/2*d*e/f)^2 - 4*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^4*tan(1/2*d*e/f)^2 + 4*s
in_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)^4*tan(1/2*d*e/f)^2 - 16*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan
(1/2*c)^3*tan(d*e/f)*tan(1/2*d*e/f)^2 + 16*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^3*tan(d*e/f)*t
an(1/2*d*e/f)^2 - 32*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)^3*tan(d*e/f)*tan(1/2*d*e/f)^2 + 6*pi*tan(1/2*c
)^2*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 12*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f)^2*tan(
1/2*d*e/f)^2 - 12*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 24*si
n_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)^2*tan(d*e/f)^2*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) - 4*real_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^4*tan(d*e/f) + 8*imag
_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^3*tan(d*e/f)^2 - 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c
)^3*tan(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 + 8*real_part(cos_inte
gral(-2*d*x - 2*d*e/f))*tan(1/2*c)^3*tan(d*e/f)^2 + 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)^3*tan(d*e/f)^2
+ 8*imag_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^4*tan(1/2*d*e/f) - 8*imag_part(cos_integral(-d*x - d*e/f)
)*tan(1/2*c)^4*tan(1/2*d*e/f) + 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)^4*tan(1/2*d*e/f) - 8*imag_part(cos
_integral(d*x + d*e/f))*tan(1/2*c)^3*tan(1/2*d*e/f)^2 + 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^3*t
an(1/2*d*e/f)^2 - 8*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^3*tan(1/2*d*e/f)^2 - 8*real_part(cos_i
ntegral(-2*d*x - 2*d*e/f))*tan(1/2*c)^3*tan(1/2*d*e/f)^2 - 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)^3*tan(1
/2*d*e/f)^2 + 24*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f)*tan(1/2*d*e/f)^2 + 24*real_p
art(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2*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)*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)*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)*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)*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 16*sin_integral((d*f*x
+ d*e)/f)*tan(1/2*c)*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 3*pi*tan(1/2*c)^4 + 2*imag_part(cos_integral(2*d*x + 2*d*
e/f))*tan(1/2*c)^4 - 2*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^4 + 4*real_part(cos_integral(d*x +
d*e/f))*tan(1/2*c)^4 + 4*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)^4 + 4*sin_integral(2*(d*f*x + d*e)/
f)*tan(1/2*c)^4 - 16*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^3*tan(d*e/f) + 16*imag_part(cos_integ
ral(-2*d*x - 2*d*e/f))*tan(1/2*c)^3*tan(d*e/f) - 32*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)^3*tan(d*e/f) +
6*pi*tan(1/2*c)^2*tan(d*e/f)^2 + 12*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f)^2 - 12*im
ag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f)^2 + 24*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2
*c)^2*tan(d*e/f)^2 - 16*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)^3*tan(1/2*d*e/f) - 16*real_part(cos_in
tegral(-d*x - d*e/f))*tan(1/2*c)^3*tan(1/2*d*e/f) - 16*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)*tan(d*e
/f)^2*tan(1/2*d*e/f) - 16*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan(d*e/f)^2*tan(1/2*d*e/f) + 6*pi*
tan(1/2*c)^2*tan(1/2*d*e/f)^2 - 12*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + 12
*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2*tan(1/2*d*e/f)^2 - 24*sin_integral(2*(d*f*x + d*e)/f)*
tan(1/2*c)^2*tan(1/2*d*e/f)^2 + 16*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)*tan(d*e/f)*tan(1/2*d*e/
f)^2 - 16*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)*tan(d*e/f)*tan(1/2*d*e/f)^2 + 32*sin_integral(2
*(d*f*x + d*e)/f)*tan(1/2*c)*tan(d*e/f)*tan(1/2*d*e/f)^2 + 3*pi*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 2*imag_part(co
s_integral(2*d*x + 2*d*e/f))*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(d
*e/f)^2*tan(1/2*d*e/f)^2 + 4*real_part(cos_integral(d*x + d*e/f))*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 4*real_part(
cos_integral(-d*x - d*e/f))*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 4*sin_integral(2*(d*f*x + d*e)/f)*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 - 8*imag_part(cos_integral(-d*x - d*e/f))*
tan(1/2*c)^3 - 8*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^3 - 8*real_part(cos_integral(-2*d*x - 2*d
*e/f))*tan(1/2*c)^3 + 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)^3 + 24*real_part(cos_integral(2*d*x + 2*d*e/
f))*tan(1/2*c)^2*tan(d*e/f) + 24*real_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2*tan(d*e/f) + 8*imag_pa
rt(cos_integral(d*x + d*e/f))*tan(1/2*c)*tan(d*e/f)^2 - 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan
(d*e/f)^2 - 8*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)*tan(d*e/f)^2 - 8*real_part(cos_integral(-2*d
*x - 2*d*e/f))*tan(1/2*c)*tan(d*e/f)^2 + 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)*tan(d*e/f)^2 - 8*imag_par
t(cos_integral(d*x + d*e/f))*tan(d*e/f)^2*tan(1/2*d*e/f) + 8*imag_part(cos_integral(-d*x - d*e/f))*tan(d*e/f)^
2*tan(1/2*d*e/f) - 16*sin_integral((d*f*x + d*e)/f)*tan(d*e/f)^2*tan(1/2*d*e/f) - 8*imag_part(cos_integral(d*x
+ d*e/f))*tan(1/2*c)*tan(1/2*d*e/f)^2 + 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan(1/2*d*e/f)^2 +
8*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)*tan(1/2*d*e/f)^2 + 8*real_part(cos_integral(-2*d*x - 2*
d*e/f))*tan(1/2*c)*tan(1/2*d*e/f)^2 - 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c)*tan(1/2*d*e/f)^2 - 4*real_pa
rt(cos_integral(2*d*x + 2*d*e/f))*tan(d*e/f)*tan(1/2*d*e/f)^2 - 4*real_part(cos_integral(-2*d*x - 2*d*e/f))*ta
n(d*e/f)*tan(1/2*d*e/f)^2 + 6*pi*tan(1/2*c)^2 - 12*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)^2 + 12*
imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c)^2 - 24*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)^2 + 16*
imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*c)*tan(d*e/f) - 16*imag_part(cos_integral(-2*d*x - 2*d*e/f))*
tan(1/2*c)*tan(d*e/f) + 32*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2*c)*tan(d*e/f) + 3*pi*tan(d*e/f)^2 - 2*imag_
part(cos_integral(2*d*x + 2*d*e/f))*tan(d*e/f)^2 + 2*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(d*e/f)^2 -
4*real_part(cos_integral(d*x + d*e/f))*tan(d*e/f)^2 - 4*real_part(cos_integral(-d*x - d*e/f))*tan(d*e/f)^2 - 4
*sin_integral(2*(d*f*x + d*e)/f)*tan(d*e/f)^2 - 16*real_part(cos_integral(d*x + d*e/f))*tan(1/2*c)*tan(1/2*d*e
/f) - 16*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*c)*tan(1/2*d*e/f) + 3*pi*tan(1/2*d*e/f)^2 + 2*imag_part
(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*d*e/f)^2 - 2*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*d*e/f)^
2 + 4*real_part(cos_integral(d*x + d*e/f))*tan(1/2*d*e/f)^2 + 4*real_part(cos_integral(-d*x - d*e/f))*tan(1/2*
d*e/f)^2 + 4*sin_integral(2*(d*f*x + d*e)/f)*tan(1/2*d*e/f)^2 + 8*imag_part(cos_integral(d*x + d*e/f))*tan(1/2
*c) - 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*c) + 8*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(1/2*
c) + 8*real_part(cos_integral(-2*d*x - 2*d*e/f))*tan(1/2*c) + 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*c) - 4*
real_part(cos_integral(2*d*x + 2*d*e/f))*tan(d*e/f) - 4*real_part(cos_integral(-2*d*x - 2*d*e/f))*tan(d*e/f) -
8*imag_part(cos_integral(d*x + d*e/f))*tan(1/2*d*e/f) + 8*imag_part(cos_integral(-d*x - d*e/f))*tan(1/2*d*e/f
) - 16*sin_integral((d*f*x + d*e)/f)*tan(1/2*d*e/f) + 2*imag_part(cos_integral(2*d*x + 2*d*e/f)) - 2*imag_part
(cos_integral(-2*d*x - 2*d*e/f)) - 4*real_part(cos_integral(d*x + d*e/f)) - 4*real_part(cos_integral(-d*x - d*
e/f)) + 4*sin_integral(2*(d*f*x + d*e)/f))/(a*f*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + a*f*tan(1/2*c)^4*
tan(d*e/f)^2 + a*f*tan(1/2*c)^4*tan(1/2*d*e/f)^2 + 2*a*f*tan(1/2*c)^2*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + a*f*tan(
1/2*c)^4 + 2*a*f*tan(1/2*c)^2*tan(d*e/f)^2 + 2*a*f*tan(1/2*c)^2*tan(1/2*d*e/f)^2 + a*f*tan(d*e/f)^2*tan(1/2*d*
e/f)^2 + 2*a*f*tan(1/2*c)^2 + a*f*tan(d*e/f)^2 + a*f*tan(1/2*d*e/f)^2 + a*f)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x
\]
[In]
int(cos(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))),x)
[Out]
int(cos(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x)