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]
-1/8*(a-a*sin(d*x+c))^4/d/(a^3+a^3*sin(d*x+c))^4
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Rubi [A] time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00,
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 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]
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 ^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*}
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Mathematica [A] time = 0.08, 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]
-1/8*Cos[c + d*x]^8/(a^8*d*(1 + Sin[c + d*x])^8)
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fricas [B] time = 0.62, size = 82, normalized size = 2.28 \[ -\frac {{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d - 4 \, {\left (a^{8} d \cos \left (d x + c\right )^{2} - 2 \, a^{8} d\right )} \sin \left (d x + c\right )} \]
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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giac [A] time = 1.40, size = 68, normalized size = 1.89 \[ \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{8}} \]
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)
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maple [A] time = 0.25, size = 55, normalized size = 1.53 \[ \frac {-\frac {3}{\left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{1+\sin \left (d x +c \right )}-\frac {2}{\left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {4}{\left (1+\sin \left (d x +c \right )\right )^{3}}}{d \,a^{8}} \]
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+1/(1+sin(d*x+c))-2/(1+sin(d*x+c))^4+4/(1+sin(d*x+c))^3)
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maxima [B] time = 0.32, size = 74, normalized size = 2.06 \[ \frac {\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )}{{\left (a^{8} \sin \left (d x + c\right )^{4} + 4 \, a^{8} \sin \left (d x + c\right )^{3} + 6 \, a^{8} \sin \left (d x + c\right )^{2} + 4 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \]
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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mupad [B] time = 0.07, size = 64, normalized size = 1.78 \[ \frac {\frac {1}{a^8\,\left (\sin \left (c+d\,x\right )+1\right )}-\frac {3}{a^8\,{\left (\sin \left (c+d\,x\right )+1\right )}^2}+\frac {4}{a^8\,{\left (\sin \left (c+d\,x\right )+1\right )}^3}-\frac {2}{a^8\,{\left (\sin \left (c+d\,x\right )+1\right )}^4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^7/(a + a*sin(c + d*x))^8,x)
[Out]
(1/(a^8*(sin(c + d*x) + 1)) - 3/(a^8*(sin(c + d*x) + 1)^2) + 4/(a^8*(sin(c + d*x) + 1)^3) - 2/(a^8*(sin(c + d*
x) + 1)^4))/d
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sympy [A] time = 42.65, size = 2006, normalized size = 55.72 \[ \text {result too large to display} \]
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((16*sin(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) + 77*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) - 8*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) + 155*sin(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) - 21*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)
+ 168*sin(c + d*x)**3/(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) - 19*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) + 104*sin(c + d*x)**2/(35*a**8*d*si
n(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 + 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)**2/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 735*a**8*d*si
n(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) + 35*sin(c + d*x)/(35*a**8*d*sin(c + d*x)**7 + 245*a**8*d*sin(c + d*x)**6 + 73
5*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) - 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) - 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) + 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), Ne(d, 0)), (x*cos(c)**7/(a*sin(c)
+ a)**8, True))
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