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

Optimal. Leaf size=257 \[ \frac {5 b \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Ci}\left (\frac {5 b c}{d}+5 b x\right )}{16 d^2}+\frac {3 b \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Ci}\left (\frac {3 b c}{d}+3 b x\right )}{16 d^2}-\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{8 d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^2}+\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{16 d^2}+\frac {5 b \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{16 d^2}-\frac {\cos (a+b x)}{8 d (c+d x)}+\frac {\cos (3 a+3 b x)}{16 d (c+d x)}+\frac {\cos (5 a+5 b x)}{16 d (c+d x)} \]

[Out]

-1/8*cos(b*x+a)/d/(d*x+c)+1/16*cos(3*b*x+3*a)/d/(d*x+c)+1/16*cos(5*b*x+5*a)/d/(d*x+c)-1/8*b*cos(a-b*c/d)*Si(b*
c/d+b*x)/d^2+3/16*b*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d^2+5/16*b*cos(5*a-5*b*c/d)*Si(5*b*c/d+5*b*x)/d^2+5/16*
b*Ci(5*b*c/d+5*b*x)*sin(5*a-5*b*c/d)/d^2+3/16*b*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d^2-1/8*b*Ci(b*c/d+b*x)*sin
(a-b*c/d)/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.34, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {5 b \sin \left (5 a-\frac {5 b c}{d}\right ) \text {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right )}{16 d^2}+\frac {3 b \sin \left (3 a-\frac {3 b c}{d}\right ) \text {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{16 d^2}-\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{8 d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^2}+\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{16 d^2}+\frac {5 b \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{16 d^2}-\frac {\cos (a+b x)}{8 d (c+d x)}+\frac {\cos (3 a+3 b x)}{16 d (c+d x)}+\frac {\cos (5 a+5 b x)}{16 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[a + b*x]/(8*d*(c + d*x)) + Cos[3*a + 3*b*x]/(16*d*(c + d*x)) + Cos[5*a + 5*b*x]/(16*d*(c + d*x)) + (5*b*C
osIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(16*d^2) + (3*b*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (
3*b*c)/d])/(16*d^2) - (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d^2) - (b*Cos[a - (b*c)/d]*SinIntegra
l[(b*c)/d + b*x])/(8*d^2) + (3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d^2) + (5*b*Cos[5*a
- (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*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 ^3(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx &=\int \left (\frac {\cos (a+b x)}{8 (c+d x)^2}-\frac {\cos (3 a+3 b x)}{16 (c+d x)^2}-\frac {\cos (5 a+5 b x)}{16 (c+d x)^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\cos (3 a+3 b x)}{(c+d x)^2} \, dx\right )-\frac {1}{16} \int \frac {\cos (5 a+5 b x)}{(c+d x)^2} \, dx+\frac {1}{8} \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx\\ &=-\frac {\cos (a+b x)}{8 d (c+d x)}+\frac {\cos (3 a+3 b x)}{16 d (c+d x)}+\frac {\cos (5 a+5 b x)}{16 d (c+d x)}-\frac {b \int \frac {\sin (a+b x)}{c+d x} \, dx}{8 d}+\frac {(3 b) \int \frac {\sin (3 a+3 b x)}{c+d x} \, dx}{16 d}+\frac {(5 b) \int \frac {\sin (5 a+5 b x)}{c+d x} \, dx}{16 d}\\ &=-\frac {\cos (a+b x)}{8 d (c+d x)}+\frac {\cos (3 a+3 b x)}{16 d (c+d x)}+\frac {\cos (5 a+5 b x)}{16 d (c+d x)}+\frac {\left (5 b \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {5 b c}{d}+5 b x\right )}{c+d x} \, dx}{16 d}+\frac {\left (3 b \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{16 d}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{8 d}+\frac {\left (5 b \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {5 b c}{d}+5 b x\right )}{c+d x} \, dx}{16 d}+\frac {\left (3 b \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{16 d}-\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{8 d}\\ &=-\frac {\cos (a+b x)}{8 d (c+d x)}+\frac {\cos (3 a+3 b x)}{16 d (c+d x)}+\frac {\cos (5 a+5 b x)}{16 d (c+d x)}+\frac {5 b \text {Ci}\left (\frac {5 b c}{d}+5 b x\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{16 d^2}+\frac {3 b \text {Ci}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{16 d^2}-\frac {b \text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{8 d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^2}+\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{16 d^2}+\frac {5 b \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{16 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 2.11, size = 212, normalized size = 0.82 \[ \frac {-2 \left (b \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (b \left (\frac {c}{d}+x\right )\right )+b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+\frac {d \cos (a+b x)}{c+d x}\right )+5 b \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Ci}\left (\frac {5 b (c+d x)}{d}\right )+3 b \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Ci}\left (\frac {3 b (c+d x)}{d}\right )+3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )+5 b \cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b (c+d x)}{d}\right )+\frac {d \cos (3 (a+b x))}{c+d x}+\frac {d \cos (5 (a+b x))}{c+d x}}{16 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*Cos[3*(a + b*x)])/(c + d*x) + (d*Cos[5*(a + b*x)])/(c + d*x) + 5*b*CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a
- (5*b*c)/d] + 3*b*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - 2*((d*Cos[a + b*x])/(c + d*x) + b*Cos
Integral[b*(c/d + x)]*Sin[a - (b*c)/d] + b*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) + 3*b*Cos[3*a - (3*b*c)/
d]*SinIntegral[(3*b*(c + d*x))/d] + 5*b*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(16*d^2)

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 339, normalized size = 1.32 \[ \frac {32 \, d \cos \left (b x + a\right )^{5} - 32 \, d \cos \left (b x + a\right )^{3} + 10 \, {\left (b d x + b c\right )} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) + 6 \, {\left (b d x + b c\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - 4 \, {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) - 2 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 5 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {5 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{32 \, {\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(32*d*cos(b*x + a)^5 - 32*d*cos(b*x + a)^3 + 10*(b*d*x + b*c)*cos(-5*(b*c - a*d)/d)*sin_integral(5*(b*d*x
+ b*c)/d) + 6*(b*d*x + b*c)*cos(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) - 4*(b*d*x + b*c)*cos(-(b*c
- a*d)/d)*sin_integral((b*d*x + b*c)/d) - 2*((b*d*x + b*c)*cos_integral((b*d*x + b*c)/d) + (b*d*x + b*c)*cos_
integral(-(b*d*x + b*c)/d))*sin(-(b*c - a*d)/d) + 3*((b*d*x + b*c)*cos_integral(3*(b*d*x + b*c)/d) + (b*d*x +
b*c)*cos_integral(-3*(b*d*x + b*c)/d))*sin(-3*(b*c - a*d)/d) + 5*((b*d*x + b*c)*cos_integral(5*(b*d*x + b*c)/d
) + (b*d*x + b*c)*cos_integral(-5*(b*d*x + b*c)/d))*sin(-5*(b*c - a*d)/d))/(d^3*x + c*d^2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.02, size = 367, normalized size = 1.43 \[ \frac {\frac {b^{2} \left (-\frac {\cos \left (b x +a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {\frac {\Si \left (b x +a +\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\Ci \left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}}{d}\right )}{8}-\frac {b^{2} \left (-\frac {5 \cos \left (5 b x +5 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {5 \left (\frac {5 \Si \left (5 b x +5 a +\frac {-5 d a +5 c b}{d}\right ) \cos \left (\frac {-5 d a +5 c b}{d}\right )}{d}-\frac {5 \Ci \left (5 b x +5 a +\frac {-5 d a +5 c b}{d}\right ) \sin \left (\frac {-5 d a +5 c b}{d}\right )}{d}\right )}{d}\right )}{80}-\frac {b^{2} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {3 \left (\frac {3 \Si \left (3 b x +3 a +\frac {-3 d a +3 c b}{d}\right ) \cos \left (\frac {-3 d a +3 c b}{d}\right )}{d}-\frac {3 \Ci \left (3 b x +3 a +\frac {-3 d a +3 c b}{d}\right ) \sin \left (\frac {-3 d a +3 c b}{d}\right )}{d}\right )}{d}\right )}{48}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/8*b^2*(-cos(b*x+a)/((b*x+a)*d-d*a+c*b)/d-(Si(b*x+a+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci(b*x+a+(-a*d+b*c
)/d)*sin((-a*d+b*c)/d)/d)/d)-1/80*b^2*(-5*cos(5*b*x+5*a)/((b*x+a)*d-d*a+c*b)/d-5*(5*Si(5*b*x+5*a+5*(-a*d+b*c)/
d)*cos(5*(-a*d+b*c)/d)/d-5*Ci(5*b*x+5*a+5*(-a*d+b*c)/d)*sin(5*(-a*d+b*c)/d)/d)/d)-1/48*b^2*(-3*cos(3*b*x+3*a)/
((b*x+a)*d-d*a+c*b)/d-3*(3*Si(3*b*x+3*a+3*(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d-3*Ci(3*b*x+3*a+3*(-a*d+b*c)/d)*s
in(3*(-a*d+b*c)/d)/d)/d))

________________________________________________________________________________________

maxima [C]  time = 0.57, size = 439, normalized size = 1.71 \[ -\frac {1073741824 \, b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - 536870912 \, b^{2} {\left (E_{2}\left (\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{2}\left (-\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 536870912 \, b^{2} {\left (E_{2}\left (\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right ) + E_{2}\left (-\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (-1073741824 i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + 1073741824 i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (536870912 i \, E_{2}\left (\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right ) - 536870912 i \, E_{2}\left (-\frac {3 i \, b c + 3 i \, {\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (536870912 i \, E_{2}\left (\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right ) - 536870912 i \, E_{2}\left (-\frac {5 i \, b c + 5 i \, {\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{17179869184 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/17179869184*(1073741824*b^2*(exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e(2, -(I*b
*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) - 536870912*b^2*(exp_integral_e(2, (3*I*b*c + 3*I*(b*x + a
)*d - 3*I*a*d)/d) + exp_integral_e(2, -(3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d))*cos(-3*(b*c - a*d)/d) - 53687
0912*b^2*(exp_integral_e(2, (5*I*b*c + 5*I*(b*x + a)*d - 5*I*a*d)/d) + exp_integral_e(2, -(5*I*b*c + 5*I*(b*x
+ a)*d - 5*I*a*d)/d))*cos(-5*(b*c - a*d)/d) + b^2*(-1073741824*I*exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*
a*d)/d) + 1073741824*I*exp_integral_e(2, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^2*(53687
0912*I*exp_integral_e(2, (3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d) - 536870912*I*exp_integral_e(2, -(3*I*b*c +
3*I*(b*x + a)*d - 3*I*a*d)/d))*sin(-3*(b*c - a*d)/d) + b^2*(536870912*I*exp_integral_e(2, (5*I*b*c + 5*I*(b*x
+ a)*d - 5*I*a*d)/d) - 536870912*I*exp_integral_e(2, -(5*I*b*c + 5*I*(b*x + a)*d - 5*I*a*d)/d))*sin(-5*(b*c -
a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________