3.28 \(\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=67 \[ \frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]

[Out]

(2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (a + a*Sin[c + d*x])^8/(8*a^5*d)

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Rubi [A]  time = 0.0656316, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (a + a*Sin[c + d*x])^8/(8*a^5*d)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{2 (a+a \sin (c+d x))^6}{3 a^3 d}-\frac{4 (a+a \sin (c+d x))^7}{7 a^4 d}+\frac{(a+a \sin (c+d x))^8}{8 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.0961315, size = 58, normalized size = 0.87 \[ -\frac{a^3 (\sin (c+d x)+1)^3 \left (21 \sin ^2(c+d x)-54 \sin (c+d x)+37\right ) \cos ^6(c+d x)}{168 d (\sin (c+d x)-1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

-(a^3*Cos[c + d*x]^6*(1 + Sin[c + d*x])^3*(37 - 54*Sin[c + d*x] + 21*Sin[c + d*x]^2))/(168*d*(-1 + Sin[c + d*x
])^3)

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Maple [B]  time = 0.042, size = 133, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24}} \right ) +3\,{a}^{3} \left ( -1/7\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2}}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+3*a^3*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*(8/3+cos(
d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-1/2*a^3*cos(d*x+c)^6+1/5*a^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*
x+c))

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Maxima [A]  time = 0.950401, size = 146, normalized size = 2.18 \begin{align*} \frac{21 \, a^{3} \sin \left (d x + c\right )^{8} + 72 \, a^{3} \sin \left (d x + c\right )^{7} + 28 \, a^{3} \sin \left (d x + c\right )^{6} - 168 \, a^{3} \sin \left (d x + c\right )^{5} - 210 \, a^{3} \sin \left (d x + c\right )^{4} + 56 \, a^{3} \sin \left (d x + c\right )^{3} + 252 \, a^{3} \sin \left (d x + c\right )^{2} + 168 \, a^{3} \sin \left (d x + c\right )}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/168*(21*a^3*sin(d*x + c)^8 + 72*a^3*sin(d*x + c)^7 + 28*a^3*sin(d*x + c)^6 - 168*a^3*sin(d*x + c)^5 - 210*a^
3*sin(d*x + c)^4 + 56*a^3*sin(d*x + c)^3 + 252*a^3*sin(d*x + c)^2 + 168*a^3*sin(d*x + c))/d

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Fricas [A]  time = 1.80867, size = 207, normalized size = 3.09 \begin{align*} \frac{21 \, a^{3} \cos \left (d x + c\right )^{8} - 112 \, a^{3} \cos \left (d x + c\right )^{6} - 8 \,{\left (9 \, a^{3} \cos \left (d x + c\right )^{6} - 6 \, a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/168*(21*a^3*cos(d*x + c)^8 - 112*a^3*cos(d*x + c)^6 - 8*(9*a^3*cos(d*x + c)^6 - 6*a^3*cos(d*x + c)^4 - 8*a^3
*cos(d*x + c)^2 - 16*a^3)*sin(d*x + c))/d

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Sympy [A]  time = 13.7934, size = 270, normalized size = 4.03 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac{8 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac{a^{3} \sin ^{6}{\left (c + d x \right )}}{2 d} + \frac{4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{8 a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac{3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{4 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac{a^{3} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \cos ^{5}{\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)**5*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((a**3*sin(c + d*x)**8/(24*d) + 8*a**3*sin(c + d*x)**7/(35*d) + a**3*sin(c + d*x)**6*cos(c + d*x)**2/
(6*d) + a**3*sin(c + d*x)**6/(2*d) + 4*a**3*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 8*a**3*sin(c + d*x)**5/(15
*d) + a**3*sin(c + d*x)**4*cos(c + d*x)**4/(4*d) + 3*a**3*sin(c + d*x)**4*cos(c + d*x)**2/(2*d) + a**3*sin(c +
d*x)**3*cos(c + d*x)**4/d + 4*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*a**3*sin(c + d*x)**2*cos(c + d*x
)**4/(2*d) + a**3*sin(c + d*x)*cos(c + d*x)**4/d, Ne(d, 0)), (x*(a*sin(c) + a)**3*cos(c)**5, True))

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Giac [B]  time = 1.20354, size = 181, normalized size = 2.7 \begin{align*} \frac{a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{5 \, a^{3} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac{25 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{33 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac{3 \, a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{a^{3} \sin \left (5 \, d x + 5 \, c\right )}{64 \, d} + \frac{17 \, a^{3} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{55 \, a^{3} \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/1024*a^3*cos(8*d*x + 8*c)/d - 5/384*a^3*cos(6*d*x + 6*c)/d - 25/256*a^3*cos(4*d*x + 4*c)/d - 33/128*a^3*cos(
2*d*x + 2*c)/d - 3/448*a^3*sin(7*d*x + 7*c)/d - 1/64*a^3*sin(5*d*x + 5*c)/d + 17/192*a^3*sin(3*d*x + 3*c)/d +
55/64*a^3*sin(d*x + c)/d