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[c + d*x])^6) + 1/(5*a^3*d*(a + a*Sin[c + d*x])^5)
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Rubi [A] time = 0.0525286, antiderivative size = 45, 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} \[ \frac{1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{3 a^2 d (a \sin (c+d x)+a)^6} \]
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 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 ^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*}
Mathematica [A] time = 0.165404, 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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Maple [A] time = 0.122, size = 33, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{8}} \left ( -{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{5\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}} \right ) } \end{align*}
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*(-1/3/(1+sin(d*x+c))^6+1/5/(1+sin(d*x+c))^5)
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Maxima [B] time = 0.957129, size = 130, normalized size = 2.89 \begin{align*} \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} \end{align*}
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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Fricas [B] time = 1.75978, size = 261, normalized size = 5.8 \begin{align*} -\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 )}} \end{align*}
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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Sympy [A] time = 53.88, size = 2020, normalized size = 44.89 \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)**3/(a+a*sin(d*x+c))**8,x)
[Out]
Piecewise((15*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) + 105*sin(c + d*x)**8/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 22
05*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)**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) + 312*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 +
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) + 105*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) + 504*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) + 315*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) + 462*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 + 3
675*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) + 525*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) + 210*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) + 525*sin(c + d*x)**3*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 7
35*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) + 315*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)
- 42*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*sin(c +
d*x) + 105*a**8*d) + 105*sin(c + d*x)*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) - 14*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 + 22
05*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) - 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 + 2
205*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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Giac [A] time = 1.15983, size = 38, normalized size = 0.84 \begin{align*} \frac{3 \, \sin \left (d x + c\right ) - 2}{15 \, a^{8} d{\left (\sin \left (d x + c\right ) + 1\right )}^{6}} \end{align*}
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)