3.463 \(\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)
Optimal. Leaf size=77 \[ \frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5} \]
[Out]
(a^2 - b^2)/(7*b^3*d*(a + b*Sin[c + d*x])^7) - a/(3*b^3*d*(a + b*Sin[c + d*x])^6) + 1/(5*b^3*d*(a + b*Sin[c +
d*x])^5)
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Rubi [A] time = 0.0715896, antiderivative size = 77, 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 =
{2668, 697} \[ \frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]
[Out]
(a^2 - b^2)/(7*b^3*d*(a + b*Sin[c + d*x])^7) - a/(3*b^3*d*(a + b*Sin[c + d*x])^6) + 1/(5*b^3*d*(a + b*Sin[c +
d*x])^5)
Rule 2668
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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 697
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a^2+b^2}{(a+x)^8}+\frac{2 a}{(a+x)^7}-\frac{1}{(a+x)^6}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5}\\ \end{align*}
Mathematica [A] time = 0.195438, size = 54, normalized size = 0.7 \[ \frac{a^2+7 a b \sin (c+d x)+21 b^2 \sin ^2(c+d x)-15 b^2}{105 b^3 d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]
[Out]
(a^2 - 15*b^2 + 7*a*b*Sin[c + d*x] + 21*b^2*Sin[c + d*x]^2)/(105*b^3*d*(a + b*Sin[c + d*x])^7)
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Maple [A] time = 0.155, size = 67, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{-{a}^{2}+{b}^{2}}{7\,{b}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{1}{5\,{b}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{a}{3\,{b}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^3/(a+b*sin(d*x+c))^8,x)
[Out]
1/d*(-1/7*(-a^2+b^2)/b^3/(a+b*sin(d*x+c))^7+1/5/b^3/(a+b*sin(d*x+c))^5-1/3*a/b^3/(a+b*sin(d*x+c))^6)
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Maxima [B] time = 0.989478, size = 204, normalized size = 2.65 \begin{align*} \frac{21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \,{\left (b^{10} \sin \left (d x + c\right )^{7} + 7 \, a b^{9} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{8} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{7} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{6} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{5} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{4} \sin \left (d x + c\right ) + a^{7} b^{3}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
[Out]
1/105*(21*b^2*sin(d*x + c)^2 + 7*a*b*sin(d*x + c) + a^2 - 15*b^2)/((b^10*sin(d*x + c)^7 + 7*a*b^9*sin(d*x + c)
^6 + 21*a^2*b^8*sin(d*x + c)^5 + 35*a^3*b^7*sin(d*x + c)^4 + 35*a^4*b^6*sin(d*x + c)^3 + 21*a^5*b^5*sin(d*x +
c)^2 + 7*a^6*b^4*sin(d*x + c) + a^7*b^3)*d)
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Fricas [B] time = 4.19691, size = 572, normalized size = 7.43 \begin{align*} \frac{21 \, b^{2} \cos \left (d x + c\right )^{2} - 7 \, a b \sin \left (d x + c\right ) - a^{2} - 6 \, b^{2}}{105 \,{\left (7 \, a b^{9} d \cos \left (d x + c\right )^{6} - 7 \,{\left (5 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{4} + 7 \,{\left (3 \, a^{5} b^{5} + 10 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{7} b^{3} + 21 \, a^{5} b^{5} + 35 \, a^{3} b^{7} + 7 \, a b^{9}\right )} d +{\left (b^{10} d \cos \left (d x + c\right )^{6} - 3 \,{\left (7 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} +{\left (35 \, a^{4} b^{6} + 42 \, a^{2} b^{8} + 3 \, b^{10}\right )} d \cos \left (d x + c\right )^{2} -{\left (7 \, a^{6} b^{4} + 35 \, a^{4} b^{6} + 21 \, a^{2} b^{8} + b^{10}\right )} 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+b*sin(d*x+c))^8,x, algorithm="fricas")
[Out]
1/105*(21*b^2*cos(d*x + c)^2 - 7*a*b*sin(d*x + c) - a^2 - 6*b^2)/(7*a*b^9*d*cos(d*x + c)^6 - 7*(5*a^3*b^7 + 3*
a*b^9)*d*cos(d*x + c)^4 + 7*(3*a^5*b^5 + 10*a^3*b^7 + 3*a*b^9)*d*cos(d*x + c)^2 - (a^7*b^3 + 21*a^5*b^5 + 35*a
^3*b^7 + 7*a*b^9)*d + (b^10*d*cos(d*x + c)^6 - 3*(7*a^2*b^8 + b^10)*d*cos(d*x + c)^4 + (35*a^4*b^6 + 42*a^2*b^
8 + 3*b^10)*d*cos(d*x + c)^2 - (7*a^6*b^4 + 35*a^4*b^6 + 21*a^2*b^8 + b^10)*d)*sin(d*x + c))
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Sympy [A] time = 63.344, size = 2632, normalized size = 34.18 \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+b*sin(d*x+c))**8,x)
[Out]
Piecewise((zoo*x*cos(c)**3/sin(c)**8, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((2/(35*d*sin(c + d*x)**5) - cos(c + d*
x)**2/(7*d*sin(c + d*x)**7))/b**8, Eq(a, 0)), ((2*sin(c + d*x)**3/(3*d) + sin(c + d*x)*cos(c + d*x)**2/d)/a**8
, Eq(b, 0)), (x*cos(c)**3/(a + b*sin(c))**8, Eq(d, 0)), (-2*a**9/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c +
d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)*
*4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) -
14*a**8*b*sin(c + d*x)/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 +
3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 73
5*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) - 42*a**7*b**2*sin(c + d*x)**2/(105*a**14*b*
*3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 +
3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*
a**7*b**10*d*sin(c + d*x)**7) + 105*a**6*b**3*sin(c + d*x)*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**4*
d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin
(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d
*x)**7) + 210*a**5*b**4*sin(c + d*x)**4/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*
sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin
(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 315*a**5*b**4*sin(c + d*x
)**2*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3
675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*
a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 462*a**4*b**5*sin(c + d*x)**5/(105*a**14*b**
3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 +
3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a
**7*b**10*d*sin(c + d*x)**7) + 525*a**4*b**5*sin(c + d*x)**3*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**
4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*s
in(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c +
d*x)**7) + 504*a**3*b**6*sin(c + d*x)**6/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*
d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*s
in(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 525*a**3*b**6*sin(c + d
*x)**4*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 +
3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 73
5*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 312*a**2*b**7*sin(c + d*x)**7/(105*a**14*b
**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3
+ 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105
*a**7*b**10*d*sin(c + d*x)**7) + 315*a**2*b**7*sin(c + d*x)**5*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b
**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d
*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c
+ d*x)**7) + 105*a*b**8*sin(c + d*x)**8/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d
*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*si
n(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 105*a*b**8*sin(c + d*x)*
*6*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 367
5*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a*
*8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7) + 15*b**9*sin(c + d*x)**9/(105*a**14*b**3*d + 73
5*a**13*b**4*d*sin(c + d*x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**
10*b**7*d*sin(c + d*x)**4 + 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**1
0*d*sin(c + d*x)**7) + 15*b**9*sin(c + d*x)**7*cos(c + d*x)**2/(105*a**14*b**3*d + 735*a**13*b**4*d*sin(c + d*
x) + 2205*a**12*b**5*d*sin(c + d*x)**2 + 3675*a**11*b**6*d*sin(c + d*x)**3 + 3675*a**10*b**7*d*sin(c + d*x)**4
+ 2205*a**9*b**8*d*sin(c + d*x)**5 + 735*a**8*b**9*d*sin(c + d*x)**6 + 105*a**7*b**10*d*sin(c + d*x)**7), Tru
e))
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Giac [A] time = 1.42951, size = 70, normalized size = 0.91 \begin{align*} \frac{21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^8,x, algorithm="giac")
[Out]
1/105*(21*b^2*sin(d*x + c)^2 + 7*a*b*sin(d*x + c) + a^2 - 15*b^2)/((b*sin(d*x + c) + a)^7*b^3*d)