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

Optimal. Leaf size=185 \[ -\frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \text{CosIntegral}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}+\frac{\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d} \]

[Out]

-(CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(16*d) + (CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b
*c)/d])/(16*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d +
b*x])/(8*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d) - (Cos[5*a - (5*b*c)/d]*SinIntegra
l[(5*b*c)/d + 5*b*x])/(16*d)

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Rubi [A]  time = 0.338659, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4406, 3303, 3299, 3302} \[ -\frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \text{CosIntegral}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}+\frac{\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(16*d) + (CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b
*c)/d])/(16*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d +
b*x])/(8*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d) - (Cos[5*a - (5*b*c)/d]*SinIntegra
l[(5*b*c)/d + 5*b*x])/(16*d)

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 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 ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{\sin (a+b x)}{8 (c+d x)}+\frac{\sin (3 a+3 b x)}{16 (c+d x)}-\frac{\sin (5 a+5 b x)}{16 (c+d x)}\right ) \, dx\\ &=\frac{1}{16} \int \frac{\sin (3 a+3 b x)}{c+d x} \, dx-\frac{1}{16} \int \frac{\sin (5 a+5 b x)}{c+d x} \, dx+\frac{1}{8} \int \frac{\sin (a+b x)}{c+d x} \, dx\\ &=-\left (\frac{1}{16} \cos \left (5 a-\frac{5 b c}{d}\right ) \int \frac{\sin \left (\frac{5 b c}{d}+5 b x\right )}{c+d x} \, dx\right )+\frac{1}{16} \cos \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{8} \cos \left (a-\frac{b c}{d}\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx-\frac{1}{16} \sin \left (5 a-\frac{5 b c}{d}\right ) \int \frac{\cos \left (\frac{5 b c}{d}+5 b x\right )}{c+d x} \, dx+\frac{1}{16} \sin \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{8} \sin \left (a-\frac{b c}{d}\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{5 b c}{d}+5 b x\right ) \sin \left (5 a-\frac{5 b c}{d}\right )}{16 d}+\frac{\text{Ci}\left (\frac{3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{16 d}+\frac{\text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.523339, size = 154, normalized size = 0.83 \[ \frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \left (-\text{CosIntegral}\left (\frac{5 b (c+d x)}{d}\right )\right )+\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b (c+d x)}{d}\right )+2 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (b \left (\frac{c}{d}+x\right )\right )+2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (b \left (\frac{c}{d}+x\right )\right )+\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b (c+d x)}{d}\right )-\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b (c+d x)}{d}\right )}{16 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.024, size = 253, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{b}{80} \left ( 5\,{\frac{1}{d}{\it Si} \left ( 5\,bx+5\,a+5\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 5\,{\frac{-ad+bc}{d}} \right ) }-5\,{\frac{1}{d}{\it Ci} \left ( 5\,bx+5\,a+5\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 5\,{\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{b}{8} \left ({\frac{1}{d}{\it Si} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \cos \left ({\frac{-ad+bc}{d}} \right ) }-{\frac{1}{d}{\it Ci} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \sin \left ({\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{b}{48} \left ( 3\,{\frac{1}{d}{\it Si} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 3\,{\frac{-ad+bc}{d}} \right ) }-3\,{\frac{1}{d}{\it Ci} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 3\,{\frac{-ad+bc}{d}} \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)^3/(d*x+c),x)

[Out]

1/b*(-1/80*b*(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)+1/8*b*(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)+1/48*
b*(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)*sin(3*(-a*d+b*c)/d)/d))

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Maxima [C]  time = 1.72738, size = 549, normalized size = 2.97 \begin{align*} \frac{b{\left (-2 i \, E_{1}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + 2 i \, E_{1}\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{\left (-i \, E_{1}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + i \, E_{1}\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{\left (i \, E_{1}\left (\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right ) - i \, E_{1}\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 ) - 2 \, b{\left (E_{1}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\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{\left (E_{1}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{1}\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{\left (E_{1}\left (\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right ) + E_{1}\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 )}{32 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(b*(-2*I*exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + 2*I*exp_integral_e(1, -(I*b*c + I*(b*x +
a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b*(-I*exp_integral_e(1, (3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d) + I*e
xp_integral_e(1, -(3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d))*cos(-3*(b*c - a*d)/d) + b*(I*exp_integral_e(1, (5*
I*b*c + 5*I*(b*x + a)*d - 5*I*a*d)/d) - I*exp_integral_e(1, -(5*I*b*c + 5*I*(b*x + a)*d - 5*I*a*d)/d))*cos(-5*
(b*c - a*d)/d) - 2*b*(exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e(1, -(I*b*c + I*(b*
x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) - b*(exp_integral_e(1, (3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d) + ex
p_integral_e(1, -(3*I*b*c + 3*I*(b*x + a)*d - 3*I*a*d)/d))*sin(-3*(b*c - a*d)/d) + b*(exp_integral_e(1, (5*I*b
*c + 5*I*(b*x + a)*d - 5*I*a*d)/d) + exp_integral_e(1, -(5*I*b*c + 5*I*(b*x + a)*d - 5*I*a*d)/d))*sin(-5*(b*c
- a*d)/d))/(b*d)

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Fricas [A]  time = 0.50882, size = 612, normalized size = 3.31 \begin{align*} \frac{2 \,{\left (\operatorname{Ci}\left (\frac{b d x + b c}{d}\right ) + \operatorname{Ci}\left (-\frac{b d x + b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right ) +{\left (\operatorname{Ci}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\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 ) -{\left (\operatorname{Ci}\left (\frac{5 \,{\left (b d x + b c\right )}}{d}\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 ) - 2 \, \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 ) + 2 \, \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 \, \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{Si}\left (\frac{b d x + b c}{d}\right )}{32 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError