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

Optimal. Leaf size=22 \[ \frac{(a \sin (c+d x)+a)^9}{9 a d} \]

[Out]

(a + a*Sin[c + d*x])^9/(9*a*d)

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Rubi [A]  time = 0.0243345, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac{(a \sin (c+d x)+a)^9}{9 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + a*Sin[c + d*x])^9/(9*a*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^8 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(a+a \sin (c+d x))^9}{9 a d}\\ \end{align*}

Mathematica [B]  time = 0.0894273, size = 147, normalized size = 6.68 \[ \frac{a^8 \sin ^9(c+d x)}{9 d}+\frac{a^8 \sin ^8(c+d x)}{d}+\frac{4 a^8 \sin ^7(c+d x)}{d}+\frac{28 a^8 \sin ^6(c+d x)}{3 d}+\frac{14 a^8 \sin ^5(c+d x)}{d}+\frac{14 a^8 \sin ^4(c+d x)}{d}+\frac{28 a^8 \sin ^3(c+d x)}{3 d}+\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{a^8 \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*Sin[c + d*x])/d + (4*a^8*Sin[c + d*x]^2)/d + (28*a^8*Sin[c + d*x]^3)/(3*d) + (14*a^8*Sin[c + d*x]^4)/d +
(14*a^8*Sin[c + d*x]^5)/d + (28*a^8*Sin[c + d*x]^6)/(3*d) + (4*a^8*Sin[c + d*x]^7)/d + (a^8*Sin[c + d*x]^8)/d
+ (a^8*Sin[c + d*x]^9)/(9*d)

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Maple [A]  time = 0.02, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{9}}{9\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a+a*sin(d*x+c))^9/d/a

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Maxima [A]  time = 0.93696, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a*sin(d*x + c) + a)^9/(a*d)

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Fricas [B]  time = 1.84925, size = 304, normalized size = 13.82 \begin{align*} \frac{9 \, a^{8} \cos \left (d x + c\right )^{8} - 120 \, a^{8} \cos \left (d x + c\right )^{6} + 432 \, a^{8} \cos \left (d x + c\right )^{4} - 576 \, a^{8} \cos \left (d x + c\right )^{2} +{\left (a^{8} \cos \left (d x + c\right )^{8} - 40 \, a^{8} \cos \left (d x + c\right )^{6} + 240 \, a^{8} \cos \left (d x + c\right )^{4} - 448 \, a^{8} \cos \left (d x + c\right )^{2} + 256 \, a^{8}\right )} \sin \left (d x + c\right )}{9 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(9*a^8*cos(d*x + c)^8 - 120*a^8*cos(d*x + c)^6 + 432*a^8*cos(d*x + c)^4 - 576*a^8*cos(d*x + c)^2 + (a^8*co
s(d*x + c)^8 - 40*a^8*cos(d*x + c)^6 + 240*a^8*cos(d*x + c)^4 - 448*a^8*cos(d*x + c)^2 + 256*a^8)*sin(d*x + c)
)/d

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Sympy [A]  time = 19.8771, size = 148, normalized size = 6.73 \begin{align*} \begin{cases} \frac{a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{a^{8} \sin ^{8}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{14 a^{8} \sin ^{5}{\left (c + d x \right )}}{d} + \frac{14 a^{8} \sin ^{4}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 a^{8} \sin ^{2}{\left (c + d x \right )}}{d} + \frac{a^{8} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{8} \cos{\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)*(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((a**8*sin(c + d*x)**9/(9*d) + a**8*sin(c + d*x)**8/d + 4*a**8*sin(c + d*x)**7/d + 28*a**8*sin(c + d*
x)**6/(3*d) + 14*a**8*sin(c + d*x)**5/d + 14*a**8*sin(c + d*x)**4/d + 28*a**8*sin(c + d*x)**3/(3*d) + 4*a**8*s
in(c + d*x)**2/d + a**8*sin(c + d*x)/d, Ne(d, 0)), (x*(a*sin(c) + a)**8*cos(c), True))

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Giac [A]  time = 1.20482, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(a*sin(d*x + c) + a)^9/(a*d)