3.78 \(\int \frac{\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx\)

Optimal. Leaf size=270 \[ -\frac{b^3 \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{24 d^4}-\frac{9 b^3 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}+\frac{b^3 \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{24 d^4}+\frac{9 b^3 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3} \]

[Out]

-(b*Cos[a + b*x])/(24*d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)^2) - (b^3*Cos[a - (b*c)/d]*CosI
ntegral[(b*c)/d + b*x])/(24*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^4) - Sin[a
+ b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[a + b*x])/(24*d^3*(c + d*x)) - Sin[3*a + 3*b*x]/(12*d*(c + d*x)^3) + (3*
b^2*Sin[3*a + 3*b*x])/(8*d^3*(c + d*x)) + (b^3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(24*d^4) + (9*b^3*
Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^4)

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Rubi [A]  time = 0.377452, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4406, 3297, 3303, 3299, 3302} \[ -\frac{b^3 \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{24 d^4}-\frac{9 b^3 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}+\frac{b^3 \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{24 d^4}+\frac{9 b^3 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^4,x]

[Out]

-(b*Cos[a + b*x])/(24*d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)^2) - (b^3*Cos[a - (b*c)/d]*CosI
ntegral[(b*c)/d + b*x])/(24*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^4) - Sin[a
+ b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[a + b*x])/(24*d^3*(c + d*x)) - Sin[3*a + 3*b*x]/(12*d*(c + d*x)^3) + (3*
b^2*Sin[3*a + 3*b*x])/(8*d^3*(c + d*x)) + (b^3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(24*d^4) + (9*b^3*
Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^4)

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx &=\int \left (\frac{\sin (a+b x)}{4 (c+d x)^4}+\frac{\sin (3 a+3 b x)}{4 (c+d x)^4}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\sin (a+b x)}{(c+d x)^4} \, dx+\frac{1}{4} \int \frac{\sin (3 a+3 b x)}{(c+d x)^4} \, dx\\ &=-\frac{\sin (a+b x)}{12 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}+\frac{b \int \frac{\cos (a+b x)}{(c+d x)^3} \, dx}{12 d}+\frac{b \int \frac{\cos (3 a+3 b x)}{(c+d x)^3} \, dx}{4 d}\\ &=-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}-\frac{b^2 \int \frac{\sin (a+b x)}{(c+d x)^2} \, dx}{24 d^2}-\frac{\left (3 b^2\right ) \int \frac{\sin (3 a+3 b x)}{(c+d x)^2} \, dx}{8 d^2}\\ &=-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac{b^3 \int \frac{\cos (a+b x)}{c+d x} \, dx}{24 d^3}-\frac{\left (9 b^3\right ) \int \frac{\cos (3 a+3 b x)}{c+d x} \, dx}{8 d^3}\\ &=-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac{\left (9 b^3 \cos \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}{8 d^3}-\frac{\left (b^3 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{24 d^3}+\frac{\left (9 b^3 \sin \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}{8 d^3}+\frac{\left (b^3 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{24 d^3}\\ &=-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{b^3 \cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{b c}{d}+b x\right )}{24 d^4}-\frac{9 b^3 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{Ci}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}-\frac{\sin (a+b x)}{12 d (c+d x)^3}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}+\frac{b^3 \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{24 d^4}+\frac{9 b^3 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^4}\\ \end{align*}

Mathematica [A]  time = 1.93298, size = 300, normalized size = 1.11 \[ -\frac{b^3 (c+d x)^3 \left (\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (b \left (\frac{c}{d}+x\right )\right )-\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (b \left (\frac{c}{d}+x\right )\right )\right )+27 b^3 (c+d x)^3 \left (\cos \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b (c+d x)}{d}\right )-\sin \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b (c+d x)}{d}\right )\right )+d \cos (b x) \left (b d \cos (a) (c+d x)-\sin (a) \left (b^2 (c+d x)^2-2 d^2\right )\right )+d \cos (3 b x) \left (3 b d \cos (3 a) (c+d x)-\sin (3 a) \left (9 b^2 (c+d x)^2-2 d^2\right )\right )-d \sin (b x) \left (\cos (a) \left (b^2 (c+d x)^2-2 d^2\right )+b d \sin (a) (c+d x)\right )-d \sin (3 b x) \left (\cos (3 a) \left (9 b^2 (c+d x)^2-2 d^2\right )+3 b d \sin (3 a) (c+d x)\right )}{24 d^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^4,x]

[Out]

-(d*Cos[b*x]*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) + d*Cos[3*b*x]*(3*b*d*(c + d*x)*Cos[3*
a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*Sin[3*a]) - d*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a])*Sin
[b*x] - d*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[3*a])*Sin[3*b*x] + b^3*(c + d*x)^3*(Cos
[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) + 27*b^3*(c + d*x)^3*(Cos[
3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]))/(24*d^
4*(c + d*x)^3)

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Maple [A]  time = 0.023, size = 381, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{{b}^{4}}{12} \left ( -{\frac{\sin \left ( 3\,bx+3\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}+{\frac{1}{d} \left ( -{\frac{3\,\cos \left ( 3\,bx+3\,a \right ) }{2\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-{\frac{3}{2\,d} \left ( -3\,{\frac{\sin \left ( 3\,bx+3\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+3\,{\frac{1}{d} \left ( 3\,{\frac{1}{d}{\it Si} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 3\,{\frac{-ad+bc}{d}} \right ) }+3\,{\frac{1}{d}{\it Ci} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 3\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) }+{\frac{{b}^{4}}{4} \left ( -{\frac{\sin \left ( bx+a \right ) }{3\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}+{\frac{1}{3\,d} \left ( -{\frac{\cos \left ( bx+a \right ) }{2\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-{\frac{1}{2\,d} \left ( -{\frac{\sin \left ( bx+a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+{\frac{1}{d} \left ({\frac{1}{d}{\it Si} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \sin \left ({\frac{-ad+bc}{d}} \right ) }+{\frac{1}{d}{\it Ci} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \cos \left ({\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^2*sin(b*x+a)/(d*x+c)^4,x)

[Out]

1/b*(1/12*b^4*(-sin(3*b*x+3*a)/((b*x+a)*d-a*d+b*c)^3/d+(-3/2*cos(3*b*x+3*a)/((b*x+a)*d-a*d+b*c)^2/d-3/2*(-3*si
n(3*b*x+3*a)/((b*x+a)*d-a*d+b*c)/d+3*(3*Si(3*b*x+3*a+3*(-a*d+b*c)/d)*sin(3*(-a*d+b*c)/d)/d+3*Ci(3*b*x+3*a+3*(-
a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d)/d)/d)/d)+1/4*b^4*(-1/3*sin(b*x+a)/((b*x+a)*d-a*d+b*c)^3/d+1/3*(-1/2*cos(b*x
+a)/((b*x+a)*d-a*d+b*c)^2/d-1/2*(-sin(b*x+a)/((b*x+a)*d-a*d+b*c)/d+(Si(b*x+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d
+Ci(b*x+a+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d)/d)/d)/d))

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Maxima [C]  time = 2.78691, size = 520, normalized size = 1.93 \begin{align*} -\frac{b^{4}{\left (i \, E_{4}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{4}\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 ) + b^{4}{\left (i \, E_{4}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) - i \, E_{4}\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 ) + b^{4}{\left (E_{4}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{4}\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^{4}{\left (E_{4}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{4}\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 )}{8 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} +{\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \,{\left (b c d^{3} - a d^{4}\right )}{\left (b x + a\right )}^{2} + 3 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}{\left (b x + a\right )}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c)^4,x, algorithm="maxima")

[Out]

-1/8*(b^4*(I*exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp_integral_e(4, -(I*b*c + I*(b*x + a)*
d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b^4*(I*exp_integral_e(4, (3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d) - I*exp
_integral_e(4, -(3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d))*cos(-3*(b*c - a*d)/d) + b^4*(exp_integral_e(4, (I*b*
c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b
^4*(exp_integral_e(4, (3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d) + exp_integral_e(4, -(3*I*b*c + 3*I*(b*x + a)*d
- 3*I*a*d)/d))*sin(-3*(b*c - a*d)/d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d
^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*b)

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Fricas [B]  time = 0.625508, size = 1231, normalized size = 4.56 \begin{align*} -\frac{24 \,{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{3} - 54 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \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 ) - 2 \,{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac{b c - a d}{d}\right ) \operatorname{Si}\left (\frac{b d x + b c}{d}\right ) - 16 \,{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) +{\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (\frac{b d x + b c}{d}\right ) +{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (-\frac{b d x + b c}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + 27 \,{\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname{Ci}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 8 \,{\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d -{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{48 \,{\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c)^4,x, algorithm="fricas")

[Out]

-1/48*(24*(b*d^3*x + b*c*d^2)*cos(b*x + a)^3 - 54*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*si
n(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) - 2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c
^3)*sin(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) - 16*(b*d^3*x + b*c*d^2)*cos(b*x + a) + ((b^3*d^3*x^3 +
3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral((b*d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*
b^3*c^2*d*x + b^3*c^3)*cos_integral(-(b*d*x + b*c)/d))*cos(-(b*c - a*d)/d) + 27*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^
2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(3*(b*d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x
+ b^3*c^3)*cos_integral(-3*(b*d*x + b*c)/d))*cos(-3*(b*c - a*d)/d) + 8*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*
c^2*d - (9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*cos(b*x + a)^2)*sin(b*x + a))/(d^7*x^3 + 3*c*d^
6*x^2 + 3*c^2*d^5*x + c^3*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**2*sin(b*x+a)/(d*x+c)**4,x)

[Out]

Integral(sin(a + b*x)*cos(a + b*x)**2/(c + d*x)**4, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*sin(b*x+a)/(d*x+c)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError