3.462 \(\int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=141 \[ -\frac {\left (a^2-b^2\right )^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac {2 a \left (a^2-b^2\right )}{3 b^5 d (a+b \sin (c+d x))^6}-\frac {2 \left (3 a^2-b^2\right )}{5 b^5 d (a+b \sin (c+d x))^5}-\frac {1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac {a}{b^5 d (a+b \sin (c+d x))^4} \]

[Out]

-1/7*(a^2-b^2)^2/b^5/d/(a+b*sin(d*x+c))^7+2/3*a*(a^2-b^2)/b^5/d/(a+b*sin(d*x+c))^6-2/5*(3*a^2-b^2)/b^5/d/(a+b*
sin(d*x+c))^5+a/b^5/d/(a+b*sin(d*x+c))^4-1/3/b^5/d/(a+b*sin(d*x+c))^3

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Rubi [A]  time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, 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} \[ -\frac {\left (a^2-b^2\right )^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac {2 a \left (a^2-b^2\right )}{3 b^5 d (a+b \sin (c+d x))^6}-\frac {2 \left (3 a^2-b^2\right )}{5 b^5 d (a+b \sin (c+d x))^5}-\frac {1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac {a}{b^5 d (a+b \sin (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^8,x]

[Out]

-(a^2 - b^2)^2/(7*b^5*d*(a + b*Sin[c + d*x])^7) + (2*a*(a^2 - b^2))/(3*b^5*d*(a + b*Sin[c + d*x])^6) - (2*(3*a
^2 - b^2))/(5*b^5*d*(a + b*Sin[c + d*x])^5) + a/(b^5*d*(a + b*Sin[c + d*x])^4) - 1/(3*b^5*d*(a + b*Sin[c + d*x
])^3)

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]

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]

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\left (a^2-b^2\right )^2}{(a+x)^8}-\frac {4 \left (a^3-a b^2\right )}{(a+x)^7}+\frac {2 \left (3 a^2-b^2\right )}{(a+x)^6}-\frac {4 a}{(a+x)^5}+\frac {1}{(a+x)^4}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {\left (a^2-b^2\right )^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac {2 a \left (a^2-b^2\right )}{3 b^5 d (a+b \sin (c+d x))^6}-\frac {2 \left (3 a^2-b^2\right )}{5 b^5 d (a+b \sin (c+d x))^5}+\frac {a}{b^5 d (a+b \sin (c+d x))^4}-\frac {1}{3 b^5 d (a+b \sin (c+d x))^3}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 107, normalized size = 0.76 \[ -\frac {a^4+21 b^2 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+7 a b \left (a^2-2 b^2\right ) \sin (c+d x)-2 a^2 b^2+35 a b^3 \sin ^3(c+d x)+35 b^4 \sin ^4(c+d x)+15 b^4}{105 b^5 d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^8,x]

[Out]

-1/105*(a^4 - 2*a^2*b^2 + 15*b^4 + 7*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 21*b^2*(a^2 - 2*b^2)*Sin[c + d*x]^2 + 35
*a*b^3*Sin[c + d*x]^3 + 35*b^4*Sin[c + d*x]^4)/(b^5*d*(a + b*Sin[c + d*x])^7)

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fricas [B]  time = 0.73, size = 309, normalized size = 2.19 \[ \frac {35 \, b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 19 \, a^{2} b^{2} + 8 \, b^{4} - 7 \, {\left (3 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (5 \, a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - 3 \, a b^{3}\right )} \sin \left (d x + c\right )}{105 \, {\left (7 \, a b^{11} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{9} + 3 \, a b^{11}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{7} + 10 \, a^{3} b^{9} + 3 \, a b^{11}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{5} + 21 \, a^{5} b^{7} + 35 \, a^{3} b^{9} + 7 \, a b^{11}\right )} d + {\left (b^{12} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{10} + b^{12}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{8} + 42 \, a^{2} b^{10} + 3 \, b^{12}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{6} + 35 \, a^{4} b^{8} + 21 \, a^{2} b^{10} + b^{12}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/105*(35*b^4*cos(d*x + c)^4 + a^4 + 19*a^2*b^2 + 8*b^4 - 7*(3*a^2*b^2 + 4*b^4)*cos(d*x + c)^2 - 7*(5*a*b^3*co
s(d*x + c)^2 - a^3*b - 3*a*b^3)*sin(d*x + c))/(7*a*b^11*d*cos(d*x + c)^6 - 7*(5*a^3*b^9 + 3*a*b^11)*d*cos(d*x
+ c)^4 + 7*(3*a^5*b^7 + 10*a^3*b^9 + 3*a*b^11)*d*cos(d*x + c)^2 - (a^7*b^5 + 21*a^5*b^7 + 35*a^3*b^9 + 7*a*b^1
1)*d + (b^12*d*cos(d*x + c)^6 - 3*(7*a^2*b^10 + b^12)*d*cos(d*x + c)^4 + (35*a^4*b^8 + 42*a^2*b^10 + 3*b^12)*d
*cos(d*x + c)^2 - (7*a^6*b^6 + 35*a^4*b^8 + 21*a^2*b^10 + b^12)*d)*sin(d*x + c))

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giac [A]  time = 6.19, size = 117, normalized size = 0.83 \[ -\frac {35 \, b^{4} \sin \left (d x + c\right )^{4} + 35 \, a b^{3} \sin \left (d x + c\right )^{3} + 21 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 42 \, b^{4} \sin \left (d x + c\right )^{2} + 7 \, a^{3} b \sin \left (d x + c\right ) - 14 \, a b^{3} \sin \left (d x + c\right ) + a^{4} - 2 \, a^{2} b^{2} + 15 \, b^{4}}{105 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{5} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/105*(35*b^4*sin(d*x + c)^4 + 35*a*b^3*sin(d*x + c)^3 + 21*a^2*b^2*sin(d*x + c)^2 - 42*b^4*sin(d*x + c)^2 +
7*a^3*b*sin(d*x + c) - 14*a*b^3*sin(d*x + c) + a^4 - 2*a^2*b^2 + 15*b^4)/((b*sin(d*x + c) + a)^7*b^5*d)

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maple [A]  time = 0.34, size = 127, normalized size = 0.90 \[ \frac {-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{7 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {1}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {6 a^{2}-2 b^{2}}{5 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {2 a \left (a^{2}-b^{2}\right )}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{6}}+\frac {a}{b^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*sin(d*x+c))^8,x)

[Out]

1/d*(-1/7*(a^4-2*a^2*b^2+b^4)/b^5/(a+b*sin(d*x+c))^7-1/3/b^5/(a+b*sin(d*x+c))^3-1/5*(6*a^2-2*b^2)/b^5/(a+b*sin
(d*x+c))^5+2/3*a*(a^2-b^2)/b^5/(a+b*sin(d*x+c))^6+a/b^5/(a+b*sin(d*x+c))^4)

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maxima [A]  time = 0.33, size = 206, normalized size = 1.46 \[ -\frac {35 \, b^{4} \sin \left (d x + c\right )^{4} + 35 \, a b^{3} \sin \left (d x + c\right )^{3} + a^{4} - 2 \, a^{2} b^{2} + 15 \, b^{4} + 21 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \sin \left (d x + c\right )^{2} + 7 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )}{105 \, {\left (b^{12} \sin \left (d x + c\right )^{7} + 7 \, a b^{11} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{10} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{9} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{8} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{7} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{6} \sin \left (d x + c\right ) + a^{7} b^{5}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^4*sin(d*x + c)^4 + 35*a*b^3*sin(d*x + c)^3 + a^4 - 2*a^2*b^2 + 15*b^4 + 21*(a^2*b^2 - 2*b^4)*sin(
d*x + c)^2 + 7*(a^3*b - 2*a*b^3)*sin(d*x + c))/((b^12*sin(d*x + c)^7 + 7*a*b^11*sin(d*x + c)^6 + 21*a^2*b^10*s
in(d*x + c)^5 + 35*a^3*b^9*sin(d*x + c)^4 + 35*a^4*b^8*sin(d*x + c)^3 + 21*a^5*b^7*sin(d*x + c)^2 + 7*a^6*b^6*
sin(d*x + c) + a^7*b^5)*d)

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mupad [B]  time = 0.14, size = 206, normalized size = 1.46 \[ -\frac {\frac {a^4-2\,a^2\,b^2+15\,b^4}{105\,b^5}+\frac {{\sin \left (c+d\,x\right )}^4}{3\,b}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (a^2-2\,b^2\right )}{5\,b^3}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3\,b^2}+\frac {a\,\sin \left (c+d\,x\right )\,\left (a^2-2\,b^2\right )}{15\,b^4}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + b*sin(c + d*x))^8,x)

[Out]

-((a^4 + 15*b^4 - 2*a^2*b^2)/(105*b^5) + sin(c + d*x)^4/(3*b) + (sin(c + d*x)^2*(a^2 - 2*b^2))/(5*b^3) + (a*si
n(c + d*x)^3)/(3*b^2) + (a*sin(c + d*x)*(a^2 - 2*b^2))/(15*b^4))/(d*(a^7 + b^7*sin(c + d*x)^7 + 7*a*b^6*sin(c
+ d*x)^6 + 21*a^5*b^2*sin(c + d*x)^2 + 35*a^4*b^3*sin(c + d*x)^3 + 35*a^3*b^4*sin(c + d*x)^4 + 21*a^2*b^5*sin(
c + d*x)^5 + 7*a^6*b*sin(c + d*x)))

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sympy [A]  time = 43.30, size = 1425, normalized size = 10.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((x*cos(c)**5/a**8, Eq(b, 0) & Eq(d, 0)), ((8*sin(c + d*x)**5/(15*d) + 4*sin(c + d*x)**3*cos(c + d*x)
**2/(3*d) + sin(c + d*x)*cos(c + d*x)**4/d)/a**8, Eq(b, 0)), (x*cos(c)**5/(a + b*sin(c))**8, Eq(d, 0)), (-a**4
/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c +
d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**
6 + 105*b**12*d*sin(c + d*x)**7) - 7*a**3*b*sin(c + d*x)/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 220
5*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**
2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 19*a**2*b**2*sin(c
+ d*x)**2/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*
d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c
+ d*x)**6 + 105*b**12*d*sin(c + d*x)**7) + 2*a**2*b**2*cos(c + d*x)**2/(105*a**7*b**5*d + 735*a**6*b**6*d*sin
(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x
)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 21*a
*b**3*sin(c + d*x)**3/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 367
5*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b
**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) + 14*a*b**3*sin(c + d*x)*cos(c + d*x)**2/(105*a**7*b**5*
d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*
a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*
sin(c + d*x)**7) - 8*b**4*sin(c + d*x)**4/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*s
in(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c
+ d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) + 12*b**4*sin(c + d*x)**2*cos(c + d*
x)**2/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*si
n(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d
*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 15*b**4*cos(c + d*x)**4/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x
) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2
205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7), True))

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