3.464 \(\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]

[Out]

-1/(7*b*d*(a + b*Sin[c + d*x])^7)

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Rubi [A]  time = 0.0269824, 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 = {2668, 32} \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*b*d*(a + b*Sin[c + d*x])^7)

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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 \frac{\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{1}{7 b d (a+b \sin (c+d x))^7}\\ \end{align*}

Mathematica [A]  time = 0.0780185, size = 22, normalized size = 1. \[ -\frac{1}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*b*d*(a + b*Sin[c + d*x])^7)

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Maple [A]  time = 0.047, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{7\,bd \left ( a+b\sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/b/d/(a+b*sin(d*x+c))^7

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Maxima [A]  time = 0.964449, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/((b*sin(d*x + c) + a)^7*b*d)

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Fricas [B]  time = 4.18221, size = 482, normalized size = 21.91 \begin{align*} \frac{1}{7 \,{\left (7 \, a b^{7} d \cos \left (d x + c\right )^{6} - 7 \,{\left (5 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{4} + 7 \,{\left (3 \, a^{5} b^{3} + 10 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{7} b + 21 \, a^{5} b^{3} + 35 \, a^{3} b^{5} + 7 \, a b^{7}\right )} d +{\left (b^{8} d \cos \left (d x + c\right )^{6} - 3 \,{\left (7 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{4} +{\left (35 \, a^{4} b^{4} + 42 \, a^{2} b^{6} + 3 \, b^{8}\right )} d \cos \left (d x + c\right )^{2} -{\left (7 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 21 \, a^{2} b^{6} + b^{8}\right )} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7/(7*a*b^7*d*cos(d*x + c)^6 - 7*(5*a^3*b^5 + 3*a*b^7)*d*cos(d*x + c)^4 + 7*(3*a^5*b^3 + 10*a^3*b^5 + 3*a*b^7
)*d*cos(d*x + c)^2 - (a^7*b + 21*a^5*b^3 + 35*a^3*b^5 + 7*a*b^7)*d + (b^8*d*cos(d*x + c)^6 - 3*(7*a^2*b^6 + b^
8)*d*cos(d*x + c)^4 + (35*a^4*b^4 + 42*a^2*b^6 + 3*b^8)*d*cos(d*x + c)^2 - (7*a^6*b^2 + 35*a^4*b^4 + 21*a^2*b^
6 + b^8)*d)*sin(d*x + c))

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Sympy [A]  time = 55.0178, size = 167, normalized size = 7.59 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{8}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{8} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{8}} & \text{for}\: d = 0 \\- \frac{1}{7 a^{7} b d + 49 a^{6} b^{2} d \sin{\left (c + d x \right )} + 147 a^{5} b^{3} d \sin ^{2}{\left (c + d x \right )} + 245 a^{4} b^{4} d \sin ^{3}{\left (c + d x \right )} + 245 a^{3} b^{5} d \sin ^{4}{\left (c + d x \right )} + 147 a^{2} b^{6} d \sin ^{5}{\left (c + d x \right )} + 49 a b^{7} d \sin ^{6}{\left (c + d x \right )} + 7 b^{8} d \sin ^{7}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((x*cos(c)/a**8, Eq(b, 0) & Eq(d, 0)), (sin(c + d*x)/(a**8*d), Eq(b, 0)), (x*cos(c)/(a + b*sin(c))**8
, Eq(d, 0)), (-1/(7*a**7*b*d + 49*a**6*b**2*d*sin(c + d*x) + 147*a**5*b**3*d*sin(c + d*x)**2 + 245*a**4*b**4*d
*sin(c + d*x)**3 + 245*a**3*b**5*d*sin(c + d*x)**4 + 147*a**2*b**6*d*sin(c + d*x)**5 + 49*a*b**7*d*sin(c + d*x
)**6 + 7*b**8*d*sin(c + d*x)**7), True))

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Giac [A]  time = 1.39745, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/((b*sin(d*x + c) + a)^7*b*d)