3.3.67 \(\int \frac {\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [267]

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

[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 = 0.20, 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 = {4619, 3384, 3380, 3383, 4491, 12} \begin {gather*} -\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} \end {gather*}

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 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*} \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 = 0.31, size = 105, normalized size = 0.82 \begin {gather*} -\frac {-2 \cos \left (c-\frac {d e}{f}\right ) \text {Ci}\left (d \left (\frac {e}{f}+x\right )\right )+\text {Ci}\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} \end {gather*}

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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Maple [A]
time = 0.18, size = 163, normalized size = 1.27

method	result	size
derivativedivides	\(\frac {\frac {\sinIntegral \left (-2 d x -2 c -\frac {2 \left (-c f +d e \right )}{f}\right ) \cos \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}+\frac {\cosineIntegral \left (2 d x +2 c +\frac {-2 c f +2 d e}{f}\right ) \sin \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}-\frac {\sinIntegral \left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}+\frac {\cosineIntegral \left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{a}\)	\(163\)
default	\(\frac {\frac {\sinIntegral \left (-2 d x -2 c -\frac {2 \left (-c f +d e \right )}{f}\right ) \cos \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}+\frac {\cosineIntegral \left (2 d x +2 c +\frac {-2 c f +2 d e}{f}\right ) \sin \left (\frac {-2 c f +2 d e}{f}\right )}{2 f}-\frac {\sinIntegral \left (-d x -c -\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{f}+\frac {\cosineIntegral \left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{f}}{a}\)	\(163\)
risch	\(-\frac {{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \expIntegral \left (1, i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a f}-\frac {{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \expIntegral \left (1, -i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{\frac {2 i \left (c f -d e \right )}{f}} \expIntegral \left (1, -2 i d x -2 i c -\frac {2 \left (-i c f +i d e \right )}{f}\right )}{4 a f}+\frac {i {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{f}} \expIntegral \left (1, 2 i d x +2 i c -\frac {2 i \left (c f -d e \right )}{f}\right )}{4 a f}\)	\(204\)

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,method=_RETURNVERBOSE)

[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] Result contains complex when optimal does not.
time = 0.38, size = 282, normalized size = 2.20 \begin {gather*} -\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} \end {gather*}

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 - 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)

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Fricas [A]
time = 0.36, size = 166, normalized size = 1.30 \begin {gather*} \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 {c f - d e}{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 (c f - d e\right )}}{f}\right ) - 2 \, \cos \left (-\frac {2 \, {\left (c f - d e\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) + 4 \, \sin \left (-\frac {c f - d e}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right )}{4 \, a f} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.61, size = 4828, normalized size = 37.72 \begin {gather*} \text {Too large to display} \end {gather*}

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(c...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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