3.85 \(\int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx\)
Optimal. Leaf size=104 \[ \frac {b \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Ci}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}-\frac {1}{8 d (c+d x)} \]
[Out]
-1/8/d/(d*x+c)+1/8*cos(4*b*x+4*a)/d/(d*x+c)+1/2*b*cos(4*a-4*b*c/d)*Si(4*b*c/d+4*b*x)/d^2+1/2*b*Ci(4*b*c/d+4*b*
x)*sin(4*a-4*b*c/d)/d^2
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Rubi [A] time = 0.17, antiderivative size = 104, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used =
{4406, 3297, 3303, 3299, 3302} \[ \frac {b \sin \left (4 a-\frac {4 b c}{d}\right ) \text {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}-\frac {1}{8 d (c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2,x]
[Out]
-1/(8*d*(c + d*x)) + Cos[4*a + 4*b*x]/(8*d*(c + d*x)) + (b*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d]
)/(2*d^2) + (b*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(2*d^2)
Rule 3297
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 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]
Rubi steps
\begin {align*} \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx &=\int \left (\frac {1}{8 (c+d x)^2}-\frac {\cos (4 a+4 b x)}{8 (c+d x)^2}\right ) \, dx\\ &=-\frac {1}{8 d (c+d x)}-\frac {1}{8} \int \frac {\cos (4 a+4 b x)}{(c+d x)^2} \, dx\\ &=-\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {b \int \frac {\sin (4 a+4 b x)}{c+d x} \, dx}{2 d}\\ &=-\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {\left (b \cos \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}{2 d}+\frac {\left (b \sin \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}{2 d}\\ &=-\frac {1}{8 d (c+d x)}+\frac {\cos (4 a+4 b x)}{8 d (c+d x)}+\frac {b \text {Ci}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{2 d^2}+\frac {b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 81, normalized size = 0.78 \[ \frac {4 b \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Ci}\left (\frac {4 b (c+d x)}{d}\right )+4 b \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )+\frac {d (\cos (4 (a+b x))-1)}{c+d x}}{8 d^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2,x]
[Out]
((d*(-1 + Cos[4*(a + b*x)]))/(c + d*x) + 4*b*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + 4*b*Cos[4*a
- (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d^2)
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fricas [A] time = 0.55, size = 138, normalized size = 1.33 \[ \frac {4 \, d \cos \left (b x + a\right )^{4} - 4 \, d \cos \left (b x + a\right )^{2} + 2 \, {\left (b d x + b c\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {4 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (d^{3} x + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")
[Out]
1/4*(4*d*cos(b*x + a)^4 - 4*d*cos(b*x + a)^2 + 2*(b*d*x + b*c)*cos(-4*(b*c - a*d)/d)*sin_integral(4*(b*d*x + b
*c)/d) + ((b*d*x + b*c)*cos_integral(4*(b*d*x + b*c)/d) + (b*d*x + b*c)*cos_integral(-4*(b*d*x + b*c)/d))*sin(
-4*(b*c - a*d)/d))/(d^3*x + c*d^2)
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giac [C] time = 0.97, size = 3218, normalized size = 30.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")
[Out]
1/4*(b*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)^2 - b*d*x*imag_part(c
os_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*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)^2 + 2*b*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*b*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*b*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)^2 - 2*b*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)^2 + b*c*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)^2 - b*c*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)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - b*d*x*imag_
part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*
tan(2*b*x)^2*tan(2*a)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 + 4*b*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) - 4*b*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) + 8*b*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*b*c*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*b*c*real_p
art(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - b*d*x*imag_part(cos_integral(4*b*x
+ 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b
*c/d)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(4
*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*
b*x)^2*tan(2*a)*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - b*
d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*sin_integral(4*(b*d*x + b*c)
/d)*tan(2*a)^2*tan(2*b*c/d)^2 + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a) + 2*b*d
*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/
d))*tan(2*b*x)^2*tan(2*a)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2 - 2*b*c*si
n_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2
*b*x)^2*tan(2*b*c/d) - 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 4*b*c*ima
g_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) - 4*b*c*imag_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*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*
tan(2*b*c/d) + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) + 2*b*d*x*real_part(co
s_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x
)^2*tan(2*b*c/d)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*c*sin_int
egral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*b*c/d)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*
a)*tan(2*b*c/d)^2 - 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + b*c*imag_part(
cos_integral(4*b*x + 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d)^2 - b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2
*a)^2*tan(2*b*c/d)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_i
ntegral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 - b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2 + 2*b*d*
x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*t
an(2*a) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a) - b*d*x*imag_part(cos_integral
(4*b*x + 4*b*c/d))*tan(2*a)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*b*d*x*sin_integ
ral(4*(b*d*x + b*c)/d)*tan(2*a)^2 - 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) -
2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*b*c/d) + 4*b*d*x*imag_part(cos_integral(4*
b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) - 4*b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d
) + 8*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*a)*tan(2*b*c/d) + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c
/d))*tan(2*a)^2*tan(2*b*c/d) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2*tan(2*b*c/d) - b*d*x
*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan
(2*b*c/d)^2 - 2*b*d*x*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(4*b*x + 4*
b*c/d))*tan(2*a)*tan(2*b*c/d)^2 - 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d)^2 + b*
c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 - b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*
b*x)^2 + 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2 + 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))
*tan(2*a) + 2*b*d*x*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a) - b*c*imag_part(cos_integral(4*b*x + 4*
b*c/d))*tan(2*a)^2 + b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)^2 - 2*b*c*sin_integral(4*(b*d*x +
b*c)/d)*tan(2*a)^2 - 2*b*d*x*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*b*d*x*real_part(cos_int
egral(-4*b*x - 4*b*c/d))*tan(2*b*c/d) + 4*b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a)*tan(2*b*c/d) -
4*b*c*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*a)*tan(2*b*c/d) + 8*b*c*sin_integral(4*(b*d*x + b*c)/d)
*tan(2*a)*tan(2*b*c/d) - b*c*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d)^2 + b*c*imag_part(cos_integ
ral(-4*b*x - 4*b*c/d))*tan(2*b*c/d)^2 - 2*b*c*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*c/d)^2 - d*tan(2*b*x)^2*
tan(2*b*c/d)^2 - 2*d*tan(2*b*x)*tan(2*a)*tan(2*b*c/d)^2 - d*tan(2*a)^2*tan(2*b*c/d)^2 + b*d*x*imag_part(cos_in
tegral(4*b*x + 4*b*c/d)) - b*d*x*imag_part(cos_integral(-4*b*x - 4*b*c/d)) + 2*b*d*x*sin_integral(4*(b*d*x + b
*c)/d) + 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*a) + 2*b*c*real_part(cos_integral(-4*b*x - 4*b*c
/d))*tan(2*a) - 2*b*c*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*c/d) - 2*b*c*real_part(cos_integral(-4*
b*x - 4*b*c/d))*tan(2*b*c/d) + b*c*imag_part(cos_integral(4*b*x + 4*b*c/d)) - b*c*imag_part(cos_integral(-4*b*
x - 4*b*c/d)) + 2*b*c*sin_integral(4*(b*d*x + b*c)/d) - d*tan(2*b*x)^2 - 2*d*tan(2*b*x)*tan(2*a) - d*tan(2*a)^
2)/(d^3*x*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*
b*x)^2*tan(2*a)^2 + d^3*x*tan(2*b*x)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*a)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2*t
an(2*a)^2 + c*d^2*tan(2*b*x)^2*tan(2*b*c/d)^2 + c*d^2*tan(2*a)^2*tan(2*b*c/d)^2 + d^3*x*tan(2*b*x)^2 + d^3*x*t
an(2*a)^2 + d^3*x*tan(2*b*c/d)^2 + c*d^2*tan(2*b*x)^2 + c*d^2*tan(2*a)^2 + c*d^2*tan(2*b*c/d)^2 + d^3*x + c*d^
2)
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maple [A] time = 0.03, size = 156, normalized size = 1.50 \[ \frac {-\frac {b^{2} \left (-\frac {4 \cos \left (4 b x +4 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {4 \left (\frac {4 \Si \left (4 b x +4 a +\frac {-4 d a +4 c b}{d}\right ) \cos \left (\frac {-4 d a +4 c b}{d}\right )}{d}-\frac {4 \Ci \left (4 b x +4 a +\frac {-4 d a +4 c b}{d}\right ) \sin \left (\frac {-4 d a +4 c b}{d}\right )}{d}\right )}{d}\right )}{32}-\frac {b^{2}}{8 \left (\left (b x +a \right ) d -d a +c b \right ) d}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x)
[Out]
1/b*(-1/32*b^2*(-4*cos(4*b*x+4*a)/((b*x+a)*d-d*a+c*b)/d-4*(4*Si(4*b*x+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b*c)/d)/
d-4*Ci(4*b*x+4*a+4*(-a*d+b*c)/d)*sin(4*(-a*d+b*c)/d)/d)/d)-1/8*b^2/((b*x+a)*d-d*a+c*b)/d)
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maxima [C] time = 0.44, size = 171, normalized size = 1.64 \[ \frac {64 \, b^{2} {\left (E_{2}\left (\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right ) + E_{2}\left (-\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - b^{2} {\left (64 i \, E_{2}\left (\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right ) - 64 i \, E_{2}\left (-\frac {4 i \, b c + 4 i \, {\left (b x + a\right )} d - 4 i \, a d}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 128 \, b^{2}}{1024 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2*sin(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")
[Out]
1/1024*(64*b^2*(exp_integral_e(2, (4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d) + exp_integral_e(2, -(4*I*b*c + 4*I
*(b*x + a)*d - 4*I*a*d)/d))*cos(-4*(b*c - a*d)/d) - b^2*(64*I*exp_integral_e(2, (4*I*b*c + 4*I*(b*x + a)*d - 4
*I*a*d)/d) - 64*I*exp_integral_e(2, -(4*I*b*c + 4*I*(b*x + a)*d - 4*I*a*d)/d))*sin(-4*(b*c - a*d)/d) - 128*b^2
)/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^2,x)
[Out]
int((cos(a + b*x)^2*sin(a + b*x)^2)/(c + d*x)^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)**2*sin(b*x+a)**2/(d*x+c)**2,x)
[Out]
Integral(sin(a + b*x)**2*cos(a + b*x)**2/(c + d*x)**2, x)
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