3.1117 \(\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=194 \[ -\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac{b \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3}{128} b x \left (8 a^2+b^2\right )-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d} \]

[Out]

(3*b*(8*a^2 + b^2)*x)/128 - (a*(2*a^2 + 61*b^2)*Cos[c + d*x]^5)/(560*d) + (3*b*(8*a^2 + b^2)*Cos[c + d*x]*Sin[
c + d*x])/(128*d) + (b*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - ((2*a^2 + 7*b^2)*Cos[c + d*x]^5*(a
+ b*Sin[c + d*x]))/(112*d) - (3*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + b*Sin[c
+ d*x])^3)/(8*d)

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Rubi [A]  time = 0.327028, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2862, 2669, 2635, 8} \[ -\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac{b \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3}{128} b x \left (8 a^2+b^2\right )-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]

[Out]

(3*b*(8*a^2 + b^2)*x)/128 - (a*(2*a^2 + 61*b^2)*Cos[c + d*x]^5)/(560*d) + (3*b*(8*a^2 + b^2)*Cos[c + d*x]*Sin[
c + d*x])/(128*d) + (b*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - ((2*a^2 + 7*b^2)*Cos[c + d*x]^5*(a
+ b*Sin[c + d*x]))/(112*d) - (3*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + b*Sin[c
+ d*x])^3)/(8*d)

Rule 2862

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*d*
m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && Gt
Q[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx &=-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{8} \int \cos ^4(c+d x) (3 b+3 a \sin (c+d x)) (a+b \sin (c+d x))^2 \, dx\\ &=-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{56} \int \cos ^4(c+d x) (a+b \sin (c+d x)) \left (27 a b+3 \left (2 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{336} \int \cos ^4(c+d x) \left (21 b \left (8 a^2+b^2\right )+3 a \left (2 a^2+61 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{16} \left (b \left (8 a^2+b^2\right )\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac{b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{64} \left (3 b \left (8 a^2+b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac{3 b \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac{1}{128} \left (3 b \left (8 a^2+b^2\right )\right ) \int 1 \, dx\\ &=\frac{3}{128} b \left (8 a^2+b^2\right ) x-\frac{a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac{3 b \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.839165, size = 189, normalized size = 0.97 \[ \frac{-280 a \left (8 a^2+9 b^2\right ) \cos (c+d x)-280 \left (4 a^3+3 a b^2\right ) \cos (3 (c+d x))+840 a^2 b \sin (2 (c+d x))-840 a^2 b \sin (4 (c+d x))-280 a^2 b \sin (6 (c+d x))+3360 a^2 b c+3360 a^2 b d x-224 a^3 \cos (5 (c+d x))+168 a b^2 \cos (5 (c+d x))+120 a b^2 \cos (7 (c+d x))-140 b^3 \sin (4 (c+d x))+\frac{35}{2} b^3 \sin (8 (c+d x))+840 b^3 c+420 b^3 d x}{17920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]

[Out]

(3360*a^2*b*c + 840*b^3*c + 3360*a^2*b*d*x + 420*b^3*d*x - 280*a*(8*a^2 + 9*b^2)*Cos[c + d*x] - 280*(4*a^3 + 3
*a*b^2)*Cos[3*(c + d*x)] - 224*a^3*Cos[5*(c + d*x)] + 168*a*b^2*Cos[5*(c + d*x)] + 120*a*b^2*Cos[7*(c + d*x)]
+ 840*a^2*b*Sin[2*(c + d*x)] - 840*a^2*b*Sin[4*(c + d*x)] - 140*b^3*Sin[4*(c + d*x)] - 280*a^2*b*Sin[6*(c + d*
x)] + (35*b^3*Sin[8*(c + d*x)])/2)/(17920*d)

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Maple [A]  time = 0.047, size = 180, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}}+3\,{a}^{2}b \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +3\,a{b}^{2} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +{b}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( dx+c \right ) }{64} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^3,x)

[Out]

1/d*(-1/5*a^3*cos(d*x+c)^5+3*a^2*b*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)
+1/16*d*x+1/16*c)+3*a*b^2*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)+b^3*(-1/8*sin(d*x+c)^3*cos(d*x+c)
^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c))

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Maxima [A]  time = 1.16054, size = 158, normalized size = 0.81 \begin{align*} -\frac{7168 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b - 3072 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a b^{2} - 35 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{3}}{35840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/35840*(7168*a^3*cos(d*x + c)^5 - 560*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^2*b - 30
72*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a*b^2 - 35*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*b^
3)/d

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Fricas [A]  time = 1.85865, size = 335, normalized size = 1.73 \begin{align*} \frac{1920 \, a b^{2} \cos \left (d x + c\right )^{7} - 896 \,{\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left (8 \, a^{2} b + b^{3}\right )} d x + 35 \,{\left (16 \, b^{3} \cos \left (d x + c\right )^{7} - 8 \,{\left (8 \, a^{2} b + 3 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (8 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4480*(1920*a*b^2*cos(d*x + c)^7 - 896*(a^3 + 3*a*b^2)*cos(d*x + c)^5 + 105*(8*a^2*b + b^3)*d*x + 35*(16*b^3*
cos(d*x + c)^7 - 8*(8*a^2*b + 3*b^3)*cos(d*x + c)^5 + 2*(8*a^2*b + b^3)*cos(d*x + c)^3 + 3*(8*a^2*b + b^3)*cos
(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 14.9345, size = 456, normalized size = 2.35 \begin{align*} \begin{cases} - \frac{a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac{3 a^{2} b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{9 a^{2} b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{9 a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{3 a^{2} b x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{3 a^{2} b \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{a^{2} b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac{3 a^{2} b \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{3 a b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{6 a b^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac{3 b^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{3 b^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{9 b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{3 b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{3 b^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{3 b^{3} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{11 b^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac{11 b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac{3 b^{3} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{3} \sin{\left (c \right )} \cos ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)*(a+b*sin(d*x+c))**3,x)

[Out]

Piecewise((-a**3*cos(c + d*x)**5/(5*d) + 3*a**2*b*x*sin(c + d*x)**6/16 + 9*a**2*b*x*sin(c + d*x)**4*cos(c + d*
x)**2/16 + 9*a**2*b*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*a**2*b*x*cos(c + d*x)**6/16 + 3*a**2*b*sin(c + d*
x)**5*cos(c + d*x)/(16*d) + a**2*b*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 3*a**2*b*sin(c + d*x)*cos(c + d*x)*
*5/(16*d) - 3*a*b**2*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 6*a*b**2*cos(c + d*x)**7/(35*d) + 3*b**3*x*sin(c
+ d*x)**8/128 + 3*b**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 9*b**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 3*
b**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*b**3*x*cos(c + d*x)**8/128 + 3*b**3*sin(c + d*x)**7*cos(c + d*x)
/(128*d) + 11*b**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 11*b**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) -
3*b**3*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sin(c))**3*sin(c)*cos(c)**4, True))

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Giac [A]  time = 1.32209, size = 248, normalized size = 1.28 \begin{align*} \frac{3 \, a b^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{b^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a^{2} b \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac{3 \, a^{2} b \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{3}{128} \,{\left (8 \, a^{2} b + b^{3}\right )} x - \frac{{\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (4 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{{\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac{{\left (6 \, a^{2} b + b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

3/448*a*b^2*cos(7*d*x + 7*c)/d + 1/1024*b^3*sin(8*d*x + 8*c)/d - 1/64*a^2*b*sin(6*d*x + 6*c)/d + 3/64*a^2*b*si
n(2*d*x + 2*c)/d + 3/128*(8*a^2*b + b^3)*x - 1/320*(4*a^3 - 3*a*b^2)*cos(5*d*x + 5*c)/d - 1/64*(4*a^3 + 3*a*b^
2)*cos(3*d*x + 3*c)/d - 1/64*(8*a^3 + 9*a*b^2)*cos(d*x + c)/d - 1/128*(6*a^2*b + b^3)*sin(4*d*x + 4*c)/d