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(d*x+c))^7

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.23, size = 33, normalized size = 1.50 \[ -\frac {1}{7 a^8 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{14}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.58, size = 108, normalized size = 4.91 \[ \frac {1}{7 \, {\left (7 \, a^{8} d \cos \left (d x + c\right )^{6} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{6} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 64 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

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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giac [A]  time = 0.49, size = 20, normalized size = 0.91 \[ -\frac {1}{7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} a d} \]

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)

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maple [A]  time = 0.10, size = 21, normalized size = 0.95 \[ -\frac {1}{7 a d \left (a +a \sin \left (d x +c \right )\right )^{7}} \]

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 = 0.31, size = 20, normalized size = 0.91 \[ -\frac {1}{7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} a d} \]

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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mupad [B]  time = 4.67, size = 18, normalized size = 0.82 \[ -\frac {1}{7\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(7*a^8*d*(sin(c + d*x) + 1)^7)

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sympy [A]  time = 42.27, size = 128, normalized size = 5.82 \[ \begin {cases} - \frac {1}{7 a^{8} d \sin ^{7}{\left (c + d x \right )} + 49 a^{8} d \sin ^{6}{\left (c + d x \right )} + 147 a^{8} d \sin ^{5}{\left (c + d x \right )} + 245 a^{8} d \sin ^{4}{\left (c + d x \right )} + 245 a^{8} d \sin ^{3}{\left (c + d x \right )} + 147 a^{8} d \sin ^{2}{\left (c + d x \right )} + 49 a^{8} d \sin {\left (c + d x \right )} + 7 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{8}} & \text {otherwise} \end {cases} \]

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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