3.93 \(\int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=45 \[ \frac {1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{3 a^2 d (a \sin (c+d x)+a)^6} \]

[Out]

-1/3/a^2/d/(a+a*sin(d*x+c))^6+1/5/a^3/d/(a+a*sin(d*x+c))^5

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Rubi [A]  time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{3 a^2 d (a \sin (c+d x)+a)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*a^2*d*(a + a*Sin[c + d*x])^6) + 1/(5*a^3*d*(a + a*Sin[c + d*x])^5)

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])

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 ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{(a+x)^7} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {2 a}{(a+x)^7}-\frac {1}{(a+x)^6}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {1}{3 a^2 d (a+a \sin (c+d x))^6}+\frac {1}{5 a^3 d (a+a \sin (c+d x))^5}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 43, normalized size = 0.96 \[ \frac {3 \sin (c+d x)-2}{15 a^8 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 + 3*Sin[c + d*x])/(15*a^8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^12)

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fricas [B]  time = 0.61, size = 105, normalized size = 2.33 \[ -\frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d - 2 \, {\left (3 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(3*sin(d*x + c) - 2)/(a^8*d*cos(d*x + c)^6 - 18*a^8*d*cos(d*x + c)^4 + 48*a^8*d*cos(d*x + c)^2 - 32*a^8*
d - 2*(3*a^8*d*cos(d*x + c)^4 - 16*a^8*d*cos(d*x + c)^2 + 16*a^8*d)*sin(d*x + c))

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giac [A]  time = 0.59, size = 28, normalized size = 0.62 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, a^{8} d {\left (\sin \left (d x + c\right ) + 1\right )}^{6}} \]

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]

1/15*(3*sin(d*x + c) - 2)/(a^8*d*(sin(d*x + c) + 1)^6)

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maple [A]  time = 0.27, size = 33, normalized size = 0.73 \[ \frac {\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}}{d \,a^{8}} \]

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*(1/5/(1+sin(d*x+c))^5-1/3/(1+sin(d*x+c))^6)

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maxima [B]  time = 0.32, size = 96, normalized size = 2.13 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} \sin \left (d x + c\right )^{6} + 6 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 20 \, a^{8} \sin \left (d x + c\right )^{3} + 15 \, a^{8} \sin \left (d x + c\right )^{2} + 6 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \]

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]

1/15*(3*sin(d*x + c) - 2)/((a^8*sin(d*x + c)^6 + 6*a^8*sin(d*x + c)^5 + 15*a^8*sin(d*x + c)^4 + 20*a^8*sin(d*x
+ c)^3 + 15*a^8*sin(d*x + c)^2 + 6*a^8*sin(d*x + c) + a^8)*d)

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mupad [B]  time = 0.10, size = 28, normalized size = 0.62 \[ \frac {3\,\sin \left (c+d\,x\right )-2}{15\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*sin(c + d*x) - 2)/(15*a^8*d*(sin(c + d*x) + 1)^6)

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sympy [A]  time = 41.54, size = 493, normalized size = 10.96 \[ \begin {cases} \frac {6 \sin ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {7 \sin {\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} - \frac {15 \cos ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {1}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{8}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3/(a+a*sin(d*x+c))**8,x)

[Out]

Piecewise((6*sin(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*
x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*s
in(c + d*x) + 105*a**8*d) + 7*sin(c + d*x)/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**
8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2
+ 735*a**8*d*sin(c + d*x) + 105*a**8*d) - 15*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c +
d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*
d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) + 1/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c +
d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8
*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d), Ne(d, 0)), (x*cos(c)**3/(a*sin(c) + a)**8, True))

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