3.43 \(\int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx\)

Optimal. Leaf size=45 \[ \frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d} \]

[Out]

(a + a*Sin[c + d*x])^10/(5*a^2*d) - (a + a*Sin[c + d*x])^11/(11*a^3*d)

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Rubi [A]  time = 0.0471371, antiderivative size = 45, 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} \[ \frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]

[Out]

(a + a*Sin[c + d*x])^10/(5*a^2*d) - (a + a*Sin[c + d*x])^11/(11*a^3*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 ^3(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^9 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac{(a+a \sin (c+d x))^{11}}{11 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.0431, size = 43, normalized size = 0.96 \[ -\frac{a^8 (5 \sin (c+d x)-6) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{20}}{55 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]

[Out]

-(a^8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^20*(-6 + 5*Sin[c + d*x]))/(55*d)

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Maple [B]  time = 0.052, size = 463, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+a*sin(d*x+c))^8,x)

[Out]

1/d*(a^8*(-1/11*sin(d*x+c)^7*cos(d*x+c)^4-7/99*sin(d*x+c)^5*cos(d*x+c)^4-5/99*sin(d*x+c)^3*cos(d*x+c)^4-1/33*s
in(d*x+c)*cos(d*x+c)^4+1/99*(2+cos(d*x+c)^2)*sin(d*x+c))+8*a^8*(-1/10*sin(d*x+c)^6*cos(d*x+c)^4-3/40*sin(d*x+c
)^4*cos(d*x+c)^4-1/20*sin(d*x+c)^2*cos(d*x+c)^4-1/40*cos(d*x+c)^4)+28*a^8*(-1/9*sin(d*x+c)^5*cos(d*x+c)^4-5/63
*sin(d*x+c)^3*cos(d*x+c)^4-1/21*sin(d*x+c)*cos(d*x+c)^4+1/63*(2+cos(d*x+c)^2)*sin(d*x+c))+56*a^8*(-1/8*sin(d*x
+c)^4*cos(d*x+c)^4-1/12*sin(d*x+c)^2*cos(d*x+c)^4-1/24*cos(d*x+c)^4)+70*a^8*(-1/7*sin(d*x+c)^3*cos(d*x+c)^4-3/
35*sin(d*x+c)*cos(d*x+c)^4+1/35*(2+cos(d*x+c)^2)*sin(d*x+c))+56*a^8*(-1/6*sin(d*x+c)^2*cos(d*x+c)^4-1/12*cos(d
*x+c)^4)+28*a^8*(-1/5*sin(d*x+c)*cos(d*x+c)^4+1/15*(2+cos(d*x+c)^2)*sin(d*x+c))-2*a^8*cos(d*x+c)^4+1/3*a^8*(2+
cos(d*x+c)^2)*sin(d*x+c))

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Maxima [B]  time = 0.949323, size = 181, normalized size = 4.02 \begin{align*} -\frac{5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/55*(5*a^8*sin(d*x + c)^11 + 44*a^8*sin(d*x + c)^10 + 165*a^8*sin(d*x + c)^9 + 330*a^8*sin(d*x + c)^8 + 330*
a^8*sin(d*x + c)^7 - 462*a^8*sin(d*x + c)^5 - 660*a^8*sin(d*x + c)^4 - 495*a^8*sin(d*x + c)^3 - 220*a^8*sin(d*
x + c)^2 - 55*a^8*sin(d*x + c))/d

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Fricas [B]  time = 1.95231, size = 352, normalized size = 7.82 \begin{align*} \frac{44 \, a^{8} \cos \left (d x + c\right )^{10} - 550 \, a^{8} \cos \left (d x + c\right )^{8} + 1760 \, a^{8} \cos \left (d x + c\right )^{6} - 1760 \, a^{8} \cos \left (d x + c\right )^{4} +{\left (5 \, a^{8} \cos \left (d x + c\right )^{10} - 190 \, a^{8} \cos \left (d x + c\right )^{8} + 1040 \, a^{8} \cos \left (d x + c\right )^{6} - 1568 \, a^{8} \cos \left (d x + c\right )^{4} + 256 \, a^{8} \cos \left (d x + c\right )^{2} + 512 \, a^{8}\right )} \sin \left (d x + c\right )}{55 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/55*(44*a^8*cos(d*x + c)^10 - 550*a^8*cos(d*x + c)^8 + 1760*a^8*cos(d*x + c)^6 - 1760*a^8*cos(d*x + c)^4 + (5
*a^8*cos(d*x + c)^10 - 190*a^8*cos(d*x + c)^8 + 1040*a^8*cos(d*x + c)^6 - 1568*a^8*cos(d*x + c)^4 + 256*a^8*co
s(d*x + c)^2 + 512*a^8)*sin(d*x + c))/d

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Sympy [A]  time = 56.5975, size = 445, normalized size = 9.89 \begin{align*} \begin{cases} \frac{2 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac{a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac{8 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} - \frac{2 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{56 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac{2 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{14 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{2 a^{8} \sin ^{4}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac{28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac{14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{a^{8} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{a^{8} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac{7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac{14 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{8} \cos ^{3}{\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)**3*(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((2*a**8*sin(c + d*x)**11/(99*d) + a**8*sin(c + d*x)**9*cos(c + d*x)**2/(9*d) + 8*a**8*sin(c + d*x)**
9/(9*d) + 4*a**8*sin(c + d*x)**7*cos(c + d*x)**2/d + 4*a**8*sin(c + d*x)**7/d - 2*a**8*sin(c + d*x)**6*cos(c +
d*x)**4/d + 14*a**8*sin(c + d*x)**5*cos(c + d*x)**2/d + 56*a**8*sin(c + d*x)**5/(15*d) - 2*a**8*sin(c + d*x)*
*4*cos(c + d*x)**6/d - 14*a**8*sin(c + d*x)**4*cos(c + d*x)**4/d + 2*a**8*sin(c + d*x)**4/d + 28*a**8*sin(c +
d*x)**3*cos(c + d*x)**2/(3*d) + 2*a**8*sin(c + d*x)**3/(3*d) - a**8*sin(c + d*x)**2*cos(c + d*x)**8/d - 28*a**
8*sin(c + d*x)**2*cos(c + d*x)**6/(3*d) - 14*a**8*sin(c + d*x)**2*cos(c + d*x)**4/d + 4*a**8*sin(c + d*x)**2*c
os(c + d*x)**2/d + a**8*sin(c + d*x)*cos(c + d*x)**2/d - a**8*cos(c + d*x)**10/(5*d) - 7*a**8*cos(c + d*x)**8/
(3*d) - 14*a**8*cos(c + d*x)**6/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)**8*cos(c)**3, True))

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Giac [B]  time = 1.2046, size = 181, normalized size = 4.02 \begin{align*} -\frac{5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/55*(5*a^8*sin(d*x + c)^11 + 44*a^8*sin(d*x + c)^10 + 165*a^8*sin(d*x + c)^9 + 330*a^8*sin(d*x + c)^8 + 330*
a^8*sin(d*x + c)^7 - 462*a^8*sin(d*x + c)^5 - 660*a^8*sin(d*x + c)^4 - 495*a^8*sin(d*x + c)^3 - 220*a^8*sin(d*
x + c)^2 - 55*a^8*sin(d*x + c))/d