3.91 \(\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)
Optimal. Leaf size=65 \[ -\frac{1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{4}{5 a^3 d (a \sin (c+d x)+a)^5} \]
[Out]
-4/(5*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(3*a^5*d*(a + a*Sin[c + d*x])^3) + 1/(d*(a^2 + a^2*Sin[c + d*x])^4)
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Rubi [A] time = 0.0586843, antiderivative size = 65, normalized size of antiderivative = 1.,
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} \[ -\frac{1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{4}{5 a^3 d (a \sin (c+d x)+a)^5} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]
[Out]
-4/(5*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(3*a^5*d*(a + a*Sin[c + d*x])^3) + 1/(d*(a^2 + a^2*Sin[c + d*x])^4)
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 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])
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{4 a^2}{(a+x)^6}-\frac{4 a}{(a+x)^5}+\frac{1}{(a+x)^4}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac{4}{5 a^3 d (a+a \sin (c+d x))^5}-\frac{1}{3 a^5 d (a+a \sin (c+d x))^3}+\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )^4}\\ \end{align*}
Mathematica [A] time = 0.123634, size = 58, normalized size = 0.89 \[ \frac{\left (5 \sin ^2(c+d x)-5 \sin (c+d x)+2\right ) \cos ^6(c+d x)}{15 a^8 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]
[Out]
(Cos[c + d*x]^6*(2 - 5*Sin[c + d*x] + 5*Sin[c + d*x]^2))/(15*a^8*d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^8)
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Maple [A] time = 0.115, size = 43, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{8}} \left ( -{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4}{5\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}+ \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-4} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x)
[Out]
1/d/a^8*(-1/3/(1+sin(d*x+c))^3-4/5/(1+sin(d*x+c))^5+1/(1+sin(d*x+c))^4)
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Maxima [A] time = 0.954115, size = 126, normalized size = 1.94 \begin{align*} -\frac{5 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{15 \,{\left (a^{8} \sin \left (d x + c\right )^{5} + 5 \, a^{8} \sin \left (d x + c\right )^{4} + 10 \, a^{8} \sin \left (d x + c\right )^{3} + 10 \, a^{8} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="maxima")
[Out]
-1/15*(5*sin(d*x + c)^2 - 5*sin(d*x + c) + 2)/((a^8*sin(d*x + c)^5 + 5*a^8*sin(d*x + c)^4 + 10*a^8*sin(d*x + c
)^3 + 10*a^8*sin(d*x + c)^2 + 5*a^8*sin(d*x + c) + a^8)*d)
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Fricas [A] time = 1.68664, size = 247, normalized size = 3.8 \begin{align*} \frac{5 \, \cos \left (d x + c\right )^{2} + 5 \, \sin \left (d x + c\right ) - 7}{15 \,{\left (5 \, a^{8} d \cos \left (d x + c\right )^{4} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d +{\left (a^{8} d \cos \left (d x + c\right )^{4} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, 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)^5/(a+a*sin(d*x+c))^8,x, algorithm="fricas")
[Out]
1/15*(5*cos(d*x + c)^2 + 5*sin(d*x + c) - 7)/(5*a^8*d*cos(d*x + c)^4 - 20*a^8*d*cos(d*x + c)^2 + 16*a^8*d + (a
^8*d*cos(d*x + c)^4 - 12*a^8*d*cos(d*x + c)^2 + 16*a^8*d)*sin(d*x + c))
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Sympy [A] time = 53.2147, size = 1658, normalized size = 25.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**5/(a+a*sin(d*x+c))**8,x)
[Out]
Piecewise((-2*sin(c + d*x)**9/(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) - 14*sin(c + d*x)**8/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 220
5*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) - 2*sin(c + d*x)**7*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) - 41*sin(c + d*x)**7/(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 + 36
75*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) - 14*sin(c + d
*x)**6*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) - 63*sin(c + d*x)**6/(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) - 42*sin(c + d*x)**5*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) - 49*sin(c + d*x)**5/(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) - 70*sin(c + d*x)**4
*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) - 15*sin(c + d*x)**4/(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) - 70*sin(c + d*x)**3*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 + 22
05*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) - 30*sin(c + d*x)**2*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) - 15*cos(c
+ d*x)**4/(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)**5/(a*sin(c) + a)**8, True))
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Giac [B] time = 1.16742, size = 185, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 170 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 282 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 170 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{15 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="giac")
[Out]
2/15*(15*tan(1/2*d*x + 1/2*c)^9 + 30*tan(1/2*d*x + 1/2*c)^8 + 140*tan(1/2*d*x + 1/2*c)^7 + 170*tan(1/2*d*x + 1
/2*c)^6 + 282*tan(1/2*d*x + 1/2*c)^5 + 170*tan(1/2*d*x + 1/2*c)^4 + 140*tan(1/2*d*x + 1/2*c)^3 + 30*tan(1/2*d*
x + 1/2*c)^2 + 15*tan(1/2*d*x + 1/2*c))/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^10)