3.89 \(\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)
Optimal. Leaf size=36 \[ -\frac{(a-a \sin (c+d x))^4}{8 d \left (a^3 \sin (c+d x)+a^3\right )^4} \]
[Out]
-(a - a*Sin[c + d*x])^4/(8*d*(a^3 + a^3*Sin[c + d*x])^4)
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Rubi [A] time = 0.0461098, antiderivative size = 36, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used =
{2667, 37} \[ -\frac{(a-a \sin (c+d x))^4}{8 d \left (a^3 \sin (c+d x)+a^3\right )^4} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^8,x]
[Out]
-(a - a*Sin[c + d*x])^4/(8*d*(a^3 + a^3*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 37
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3}{(a+x)^5} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac{(a-a \sin (c+d x))^4}{8 d \left (a^3+a^3 \sin (c+d x)\right )^4}\\ \end{align*}
Mathematica [A] time = 0.102754, size = 28, normalized size = 0.78 \[ -\frac{\cos ^8(c+d x)}{8 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^8,x]
[Out]
-Cos[c + d*x]^8/(8*a^8*d*(1 + Sin[c + d*x])^8)
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Maple [A] time = 0.105, size = 55, normalized size = 1.5 \begin{align*}{\frac{1}{d{a}^{8}} \left ( -3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-2}+4\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-3}+ \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-1}-2\, \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)^7/(a+a*sin(d*x+c))^8,x)
[Out]
1/d/a^8*(-3/(1+sin(d*x+c))^2+4/(1+sin(d*x+c))^3+1/(1+sin(d*x+c))-2/(1+sin(d*x+c))^4)
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Maxima [B] time = 0.956296, size = 100, normalized size = 2.78 \begin{align*} \frac{\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )}{{\left (a^{8} \sin \left (d x + c\right )^{4} + 4 \, a^{8} \sin \left (d x + c\right )^{3} + 6 \, a^{8} \sin \left (d x + c\right )^{2} + 4 \, 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)^7/(a+a*sin(d*x+c))^8,x, algorithm="maxima")
[Out]
(sin(d*x + c)^3 + sin(d*x + c))/((a^8*sin(d*x + c)^4 + 4*a^8*sin(d*x + c)^3 + 6*a^8*sin(d*x + c)^2 + 4*a^8*sin
(d*x + c) + a^8)*d)
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Fricas [B] time = 1.65309, size = 194, normalized size = 5.39 \begin{align*} -\frac{{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d - 4 \,{\left (a^{8} d \cos \left (d x + c\right )^{2} - 2 \, a^{8} d\right )} \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^8,x, algorithm="fricas")
[Out]
-(cos(d*x + c)^2 - 2)*sin(d*x + c)/(a^8*d*cos(d*x + c)^4 - 8*a^8*d*cos(d*x + c)^2 + 8*a^8*d - 4*(a^8*d*cos(d*x
+ c)^2 - 2*a^8*d)*sin(d*x + c))
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Sympy [A] time = 54.3133, size = 2032, normalized size = 56.44 \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)**7/(a+a*sin(d*x+c))**8,x)
[Out]
Piecewise((sin(c + d*x)**9/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**
5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c
+ d*x) + 35*a**8*d) + 7*sin(c + d*x)**8/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*s
in(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*
a**8*d*sin(c + d*x) + 35*a**8*d) + sin(c + d*x)**7*cos(c + d*x)**2/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin
(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a*
*8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 16*sin(c + d*x)**7/(35*a**8*d*sin(c + d*x)**7 +
245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*
x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 7*sin(c + d*x)**6*cos(c + d*x)**2/
(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x
)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 16*si
n(c + d*x)**6/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8
*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a
**8*d) + 21*sin(c + d*x)**5*cos(c + d*x)**2/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8
*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 +
245*a**8*d*sin(c + d*x) + 35*a**8*d) + 7*sin(c + d*x)**5/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)*
*6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c
+ d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 27*sin(c + d*x)**4*cos(c + d*x)**2/(35*a**8*d*sin(c + d*x)
**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(
c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + sin(c + d*x)**4/(35*a**8*d*s
in(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*
a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 14*sin(c + d*x)**
3*cos(c + d*x)**2/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*
a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) +
35*a**8*d) + 6*sin(c + d*x)**2*cos(c + d*x)**4/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a
**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2
+ 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 2*sin(c + d*x)**2*cos(c + d*x)**2/(35*a**8*d*sin(c + d*x)**7 + 245*a
**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3
+ 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) + 7*sin(c + d*x)*cos(c + d*x)**4/(35*a**8
*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c + d*x)**4 + 1
225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) - 5*cos(c + d*x
)**6/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a**8*d*sin(c
+ d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35*a**8*d) +
cos(c + d*x)**4/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*sin(c + d*x)**5 + 1225*a*
*8*d*sin(c + d*x)**4 + 1225*a**8*d*sin(c + d*x)**3 + 735*a**8*d*sin(c + d*x)**2 + 245*a**8*d*sin(c + d*x) + 35
*a**8*d), Ne(d, 0)), (x*cos(c)**7/(a*sin(c) + a)**8, True))
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Giac [A] time = 1.18639, size = 92, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7/(a+a*sin(d*x+c))^8,x, algorithm="giac")
[Out]
2*(tan(1/2*d*x + 1/2*c)^7 + 7*tan(1/2*d*x + 1/2*c)^5 + 7*tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))/(a^8*d
*(tan(1/2*d*x + 1/2*c) + 1)^8)