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

Optimal. Leaf size=128 \[ -\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \text {Ci}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Ci}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \]

[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

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Rubi [A]  time = 0.30, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4523, 3303, 3299, 3302, 4406, 12} \[ -\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \text {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \]

Antiderivative was successfully verified.

[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 3299

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 3302

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 3303

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 4406

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 4523

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*} \int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\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 ) \text {Ci}\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 ) \text {Ci}\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 ) \text {Ci}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\text {Ci}\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*}

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Mathematica [A]  time = 0.40, size = 105, normalized size = 0.82 \[ -\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \text {Ci}\left (\frac {2 d (e+f x)}{f}\right )-2 \cos \left (c-\frac {d e}{f}\right ) \text {Ci}\left (d \left (\frac {e}{f}+x\right )\right )+2 \sin \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+\cos \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.44, size = 157, normalized size = 1.23 \[ \frac {2 \, {\left (\operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) + \operatorname {Ci}\left (-\frac {d f x + d e}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - {\left (\operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) - 2 \, \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) - 4 \, \sin \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right )}{4 \, a f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(cos_integral((d*f*x + d*e)/f) + cos_integral(-(d*f*x + d*e)/f))*cos(-(d*e - c*f)/f) - (cos_integral(2*
(d*f*x + d*e)/f) + cos_integral(-2*(d*f*x + d*e)/f))*sin(-2*(d*e - c*f)/f) - 2*cos(-2*(d*e - c*f)/f)*sin_integ
ral(2*(d*f*x + d*e)/f) - 4*sin(-(d*e - c*f)/f)*sin_integral((d*f*x + d*e)/f))/(a*f)

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giac [C]  time = 4.17, size = 4828, normalized size = 37.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.11, size = 161, normalized size = 1.26 \[ -\frac {\frac {\Si \left (2 d x +2 c +\frac {-2 c f +2 d e}{f}\right ) \cos \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}-\frac {\Ci \left (2 d x +2 c +\frac {-2 c f +2 d e}{f}\right ) \sin \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}-\frac {\Si \left (d x +c +\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}-\frac {\Ci \left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.82, size = 280, normalized size = 2.19 \[ -\frac {2 \, d {\left (E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - d {\left (i \, E_{1}\left (\frac {2 i \, d e + 2 i \, {\left (d x + c\right )} f - 2 i \, c f}{f}\right ) - i \, E_{1}\left (-\frac {2 i \, d e + 2 i \, {\left (d x + c\right )} f - 2 i \, c f}{f}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) - d {\left (2 i \, E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) - 2 i \, E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - d {\left (E_{1}\left (\frac {2 i \, d e + 2 i \, {\left (d x + c\right )} f - 2 i \, c f}{f}\right ) + E_{1}\left (-\frac {2 i \, d e + 2 i \, {\left (d x + c\right )} f - 2 i \, c f}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}{4 \, a d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 + 2*I*(d*x + c)*f - 2*I*c*f)/f) - I*exp_integ
ral_e(1, -(2*I*d*e + 2*I*(d*x + c)*f - 2*I*c*f)/f))*cos(-2*(d*e - c*f)/f) - d*(2*I*exp_integral_e(1, (I*d*e +
I*(d*x + c)*f - I*c*f)/f) - 2*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 + 2*I*(d*x + c)*f - 2*I*c*f)/f) + exp_integral_e(1, -(2*I*d*e + 2*I*(d*x + c)*f -
2*I*c*f)/f))*sin(-2*(d*e - c*f)/f))/(a*d*f)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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