\(\int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx\) [86]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=-\frac {1}{16 d (c+d x)^2}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}+\frac {b^2 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{d^3}-\frac {b \sin (4 a+4 b x)}{4 d^2 (c+d x)}-\frac {b^2 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{d^3} \]

[Out]

-1/16/d/(d*x+c)^2+b^2*Ci(4*b*c/d+4*b*x)*cos(4*a-4*b*c/d)/d^3+1/16*cos(4*b*x+4*a)/d/(d*x+c)^2-b^2*Si(4*b*c/d+4*
b*x)*sin(4*a-4*b*c/d)/d^3-1/4*b*sin(4*b*x+4*a)/d^2/(d*x+c)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4491, 3378, 3384, 3380, 3383} \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{d^3}-\frac {b^2 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{d^3}-\frac {b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac {1}{16 d (c+d x)^2} \]

[In]

Int[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^3,x]

[Out]

-1/16*1/(d*(c + d*x)^2) + Cos[4*a + 4*b*x]/(16*d*(c + d*x)^2) + (b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/
d + 4*b*x])/d^3 - (b*Sin[4*a + 4*b*x])/(4*d^2*(c + d*x)) - (b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4
*b*x])/d^3

Rule 3378

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] - Dist[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 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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8 (c+d x)^3}-\frac {\cos (4 a+4 b x)}{8 (c+d x)^3}\right ) \, dx \\ & = -\frac {1}{16 d (c+d x)^2}-\frac {1}{8} \int \frac {\cos (4 a+4 b x)}{(c+d x)^3} \, dx \\ & = -\frac {1}{16 d (c+d x)^2}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}+\frac {b \int \frac {\sin (4 a+4 b x)}{(c+d x)^2} \, dx}{4 d} \\ & = -\frac {1}{16 d (c+d x)^2}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac {b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac {b^2 \int \frac {\cos (4 a+4 b x)}{c+d x} \, dx}{d^2} \\ & = -\frac {1}{16 d (c+d x)^2}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac {b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac {\left (b^2 \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx}{d^2}-\frac {\left (b^2 \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx}{d^2} \\ & = -\frac {1}{16 d (c+d x)^2}+\frac {\cos (4 a+4 b x)}{16 d (c+d x)^2}+\frac {b^2 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{d^3}-\frac {b \sin (4 a+4 b x)}{4 d^2 (c+d x)}-\frac {b^2 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\frac {16 b^2 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right )+\frac {d (-d+d \cos (4 (a+b x))-4 b (c+d x) \sin (4 (a+b x)))}{(c+d x)^2}-16 b^2 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{16 d^3} \]

[In]

Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^3,x]

[Out]

(16*b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] + (d*(-d + d*Cos[4*(a + b*x)] - 4*b*(c + d*x)*Sin[
4*(a + b*x)]))/(c + d*x)^2 - 16*b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(16*d^3)

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {-\frac {b^{3} \left (-\frac {2 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {4 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {16 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {16 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}}{d}\right )}{d}\right )}{32}-\frac {b^{3}}{16 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}}{b}\)	\(193\)
default	\(\frac {-\frac {b^{3} \left (-\frac {2 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {4 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {16 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {16 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}}{d}\right )}{d}\right )}{32}-\frac {b^{3}}{16 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}}{b}\)	\(193\)
risch	\(-\frac {1}{16 d \left (d x +c \right )^{2}}-\frac {b^{2} {\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{2 d^{3}}-\frac {b^{2} {\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{2 d^{3}}-\frac {\left (-2 b^{2} d^{3} x^{2}-4 b^{2} c \,d^{2} x -2 b^{2} c^{2} d \right ) \cos \left (4 x b +4 a \right )}{32 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}+\frac {i \left (8 i b^{3} d^{3} x^{3}+24 i b^{3} c \,d^{2} x^{2}+24 i b^{3} c^{2} d x +8 i c^{3} b^{3}\right ) \sin \left (4 x b +4 a \right )}{32 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}\)	\(290\)

[In]

int(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/32*b^3*(-2*cos(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))^2/d-2*(-4*sin(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))/d+4*(-4*S
i(-4*x*b-4*a-4*(-a*d+b*c)/d)*sin(4*(-a*d+b*c)/d)/d+4*Ci(4*x*b+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/d)/d)/d)
-1/16*b^3/(-a*d+c*b+d*(b*x+a))^2/d)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\frac {d^{2} \cos \left (b x + a\right )^{4} - d^{2} \cos \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(d^2*cos(b*x + a)^4 - d^2*cos(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(-4*(b*c - a*d)/d)*c
os_integral(4*(b*d*x + b*c)/d) - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(-4*(b*c - a*d)/d)*sin_integral(4*
(b*d*x + b*c)/d) - 2*(2*(b*d^2*x + b*c*d)*cos(b*x + a)^3 - (b*d^2*x + b*c*d)*cos(b*x + a))*sin(b*x + a))/(d^5*
x^2 + 2*c*d^4*x + c^2*d^3)

Sympy [F]

\[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]

[In]

integrate(cos(b*x+a)**2*sin(b*x+a)**2/(d*x+c)**3,x)

[Out]

Integral(sin(a + b*x)**2*cos(a + b*x)**2/(c + d*x)**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.62 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\frac {b^{3} {\left (E_{3}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + b^{3} {\left (i \, E_{3}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{3}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - b^{3}}{16 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + {\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} b} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/16*(b^3*(exp_integral_e(3, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(3, -4*(-I*b*c - I*(b*x + a
)*d + I*a*d)/d))*cos(-4*(b*c - a*d)/d) + b^3*(I*exp_integral_e(3, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*ex
p_integral_e(3, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-4*(b*c - a*d)/d) - b^3)/((b^2*c^2*d - 2*a*b*c*d^2
+ (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*b)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.49 (sec) , antiderivative size = 5600, normalized size of antiderivative = 44.09 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \]

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

1/8*(4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2
*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_par
t(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*
b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*
tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2
+ 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*
b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d)
)*tan(2*b*x)^2*tan(2*a)^2 - 4*b^2*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 +
16*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 16*b^2*d^2*x^2*re
al_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integr
al(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b*c
/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)
^2*tan(2*b*c/d) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 4*b^2*d
^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 16*b^2*c*d*x*imag_part(cos_inte
gral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b
*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*
a)*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d
^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(
4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))
*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*
tan(2*a) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 16*b^2*d^2*x^2*sin_
integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*b*x)^2*tan(2*a)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 8*b^2*d^2
*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(cos_integral
(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*ta
n(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 32*b^2
*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d
))*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d) - 8*b^2*c^
2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_inte
gral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(
2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c
/d)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + 8*b^2*d^2*x^2*imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4
*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*d^2*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d)^2 + 8*
b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 8*b^2*c^2*imag_part(co
s_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d
)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2
*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*re
al_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*
tan(2*b*x)^2 - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 16*b^2*c*d*x*imag
_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan
(2*b*x)^2*tan(2*a) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2 - 4*b^2*d^2*x^2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^
2*tan(2*a)^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + 16*b^2*c*d*x*imag
_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x -
4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d) +
16*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*d^2*x^2*real_part(cos_
integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
b*x)^2*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(
2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 16*b^2*c*d*x*imag_p
art(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan
(2*a)^2*tan(2*b*c/d) - 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*d^2*x^2*r
eal_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*t
an(2*b*x)^2*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 +
16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 16*b^2*c*d*x*imag_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(
2*b*c/d)^2 + 4*b*d^2*x*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d
))*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 +
4*b*d^2*x*tan(2*b*x)*tan(2*a)^2*tan(2*b*c/d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b
*x)^2 + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 - 8*b^2*d^2*x^2*imag_part(cos_integ
ral(4*b*x + 4*b*c/d))*tan(2*a) + 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 16*b^2*d^2
*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a) - 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^
2*tan(2*a) + 8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - 16*b^2*c^2*sin_integr
al(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a) - 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2
- 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 + 8*b^2*d^2*x^2*imag_part(cos_integral(4*b*
x + 4*b*c/d))*tan(2*b*c/d) - 8*b^2*d^2*x^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 16*b^2*d^2
*x^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b
*x)^2*tan(2*b*c/d) - 8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 16*b^2*c^
2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*
c/d))*tan(2*a)*tan(2*b*c/d) + 32*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) - 8
*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 8*b^2*c^2*imag_part(cos_integral(-
4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)
- 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 8*b^2*c*d*x*real_part(cos_integral(-4
*b*x - 4*b*c/d))*tan(2*b*c/d)^2 + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 -
8*b^2*c^2*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + 16*b^2*c^2*sin_integral(4*(b*d*
x + b*c)/d)*tan(2*a)*tan(2*b*c/d)^2 + 4*b*c*d*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + 4*b*c*d*tan(2*b*x)*tan(2*
a)^2*tan(2*b*c/d)^2 + 4*b^2*d^2*x^2*real_part(cos_integral(4*b*x + 4*b*c/d)) + 4*b^2*d^2*x^2*real_part(cos_int
egral(-4*b*x - 4*b*c/d)) + 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 + 4*b^2*c^2*real_pa
rt(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 - 16*b^2*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
a) + 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*
c)/d)*tan(2*a) + 4*b*d^2*x*tan(2*b*x)^2*tan(2*a) - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)
^2 - 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 + 4*b*d^2*x*tan(2*b*x)*tan(2*a)^2 + 16*b^2
*c*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 16*b^2*c*d*x*imag_part(cos_integral(-4*b*x - 4*
b*c/d))*tan(2*b*c/d) + 32*b^2*c*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_in
tegral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 16*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)
*tan(2*b*c/d) - 4*b^2*c^2*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 - 4*b^2*c^2*real_part(cos_in
tegral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 4*b*d^2*x*tan(2*b*x)*tan(2*b*c/d)^2 - 4*b*d^2*x*tan(2*a)*tan(2*b*c/
d)^2 + 8*b^2*c*d*x*real_part(cos_integral(4*b*x + 4*b*c/d)) + 8*b^2*c*d*x*real_part(cos_integral(-4*b*x - 4*b*
c/d)) - 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 8*b^2*c^2*imag_part(cos_integral(-4*b*x
- 4*b*c/d))*tan(2*a) - 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a) + 4*b*c*d*tan(2*b*x)^2*tan(2*a) + 4
*b*c*d*tan(2*b*x)*tan(2*a)^2 + 8*b^2*c^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 8*b^2*c^2*ima
g_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 16*b^2*c^2*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)
- 4*b*c*d*tan(2*b*x)*tan(2*b*c/d)^2 - d^2*tan(2*b*x)^2*tan(2*b*c/d)^2 - 4*b*c*d*tan(2*a)*tan(2*b*c/d)^2 - 2*d^
2*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - d^2*tan(2*a)^2*tan(2*b*c/d)^2 + 4*b^2*c^2*real_part(cos_integral(4*b*x
+ 4*b*c/d)) + 4*b^2*c^2*real_part(cos_integral(-4*b*x - 4*b*c/d)) - 4*b*d^2*x*tan(2*b*x) - 4*b*d^2*x*tan(2*a)
- 4*b*c*d*tan(2*b*x) - d^2*tan(2*b*x)^2 - 4*b*c*d*tan(2*a) - 2*d^2*tan(2*b*x)*tan(2*a) - d^2*tan(2*a)^2)/(d^5*
x^2*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*
b*x)^2*tan(2*a)^2 + d^5*x^2*tan(2*b*x)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*a)^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*b*
x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*a)^2 + 2*c*d^4*x*tan(2*b*x)^2*tan(2*b*c/d)^2 + 2
*c*d^4*x*tan(2*a)^2*tan(2*b*c/d)^2 + d^5*x^2*tan(2*b*x)^2 + d^5*x^2*tan(2*a)^2 + c^2*d^3*tan(2*b*x)^2*tan(2*a)
^2 + d^5*x^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*b*x)^2*tan(2*b*c/d)^2 + c^2*d^3*tan(2*a)^2*tan(2*b*c/d)^2 + 2*c*d^
4*x*tan(2*b*x)^2 + 2*c*d^4*x*tan(2*a)^2 + 2*c*d^4*x*tan(2*b*c/d)^2 + d^5*x^2 + c^2*d^3*tan(2*b*x)^2 + c^2*d^3*
tan(2*a)^2 + c^2*d^3*tan(2*b*c/d)^2 + 2*c*d^4*x + c^2*d^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]

[In]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^3,x)

[Out]

int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^3, x)