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

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

[Out]

-1/(7*a*d*(a + a*Sin[c + d*x])^7)

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Rubi [A]  time = 0.0250511, 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} \[ -\frac{1}{7 a d (a \sin (c+d x)+a)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*a*d*(a + a*Sin[c + d*x])^7)

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

Mathematica [A]  time = 0.228168, size = 33, normalized size = 1.5 \[ -\frac{1}{7 a^8 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{14}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*a^8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^14)

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Maple [A]  time = 0.02, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{7\,da \left ( a+a\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+a*sin(d*x+c))^8,x)

[Out]

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

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Maxima [A]  time = 0.965244, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{7} 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]

-1/7/((a*sin(d*x + c) + a)^7*a*d)

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Fricas [B]  time = 1.72512, size = 263, normalized size = 11.95 \begin{align*} \frac{1}{7 \,{\left (7 \, a^{8} d \cos \left (d x + c\right )^{6} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} d +{\left (a^{8} d \cos \left (d x + c\right )^{6} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} 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+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/7/(7*a^8*d*cos(d*x + c)^6 - 56*a^8*d*cos(d*x + c)^4 + 112*a^8*d*cos(d*x + c)^2 - 64*a^8*d + (a^8*d*cos(d*x +
c)^6 - 24*a^8*d*cos(d*x + c)^4 + 80*a^8*d*cos(d*x + c)^2 - 64*a^8*d)*sin(d*x + c))

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Sympy [A]  time = 52.7169, size = 128, normalized size = 5.82 \begin{align*} \begin{cases} - \frac{1}{7 a^{8} d \sin ^{7}{\left (c + d x \right )} + 49 a^{8} d \sin ^{6}{\left (c + d x \right )} + 147 a^{8} d \sin ^{5}{\left (c + d x \right )} + 245 a^{8} d \sin ^{4}{\left (c + d x \right )} + 245 a^{8} d \sin ^{3}{\left (c + d x \right )} + 147 a^{8} d \sin ^{2}{\left (c + d x \right )} + 49 a^{8} d \sin{\left (c + d x \right )} + 7 a^{8} d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{8}} & \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((-1/(7*a**8*d*sin(c + d*x)**7 + 49*a**8*d*sin(c + d*x)**6 + 147*a**8*d*sin(c + d*x)**5 + 245*a**8*d*
sin(c + d*x)**4 + 245*a**8*d*sin(c + d*x)**3 + 147*a**8*d*sin(c + d*x)**2 + 49*a**8*d*sin(c + d*x) + 7*a**8*d)
, Ne(d, 0)), (x*cos(c)/(a*sin(c) + a)**8, True))

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Giac [A]  time = 1.15364, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{7 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{7} 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]

-1/7/((a*sin(d*x + c) + a)^7*a*d)