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]

-(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)

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Rubi [A]  time = 0.108766, antiderivative size = 141, 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} \[ -\frac{\left (a^2-b^2\right )^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac{2 a \left (a^2-b^2\right )}{3 b^5 d (a+b \sin (c+d x))^6}-\frac{2 \left (3 a^2-b^2\right )}{5 b^5 d (a+b \sin (c+d x))^5}-\frac{1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac{a}{b^5 d (a+b \sin (c+d x))^4} \]

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 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 ^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*}

Mathematica [A]  time = 0.285082, size = 107, normalized size = 0.76 \[ -\frac{21 b^2 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+7 a b \left (a^2-2 b^2\right ) \sin (c+d x)-2 a^2 b^2+a^4+35 a b^3 \sin ^3(c+d x)+35 b^4 \sin ^4(c+d x)+15 b^4}{105 b^5 d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)/(105*b^5*d*(a + b*Sin[c + d*x])^7)

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Maple [A]  time = 0.159, size = 127, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{{a}^{4}-2\,{a}^{2}{b}^{2}+{b}^{4}}{7\,{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{6\,{a}^{2}-2\,{b}^{2}}{5\,{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{3\,{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,a \left ({a}^{2}-{b}^{2} \right ) }{3\,{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a}{{b}^{5} \left ( a+b\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)^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))^7-1/5*(6*a^2-2*b^2)/b^5/(a+b*sin(d*x+c))^5-1/3/b^5/(a+b*sin
(d*x+c))^3+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 = 1.00109, size = 278, normalized size = 1.97 \begin{align*} -\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} \end{align*}

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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Fricas [B]  time = 4.10491, size = 705, normalized size = 5. \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 65.6567, size = 2181, normalized size = 15.47 \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)**5/(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((zoo*x*cos(c)**5/sin(c)**8, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-8/(105*d*sin(c + d*x)**3) + 4*cos(c
+ d*x)**2/(35*d*sin(c + d*x)**5) - cos(c + d*x)**4/(7*d*sin(c + d*x)**7))/b**8, Eq(a, 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)), (-15*a**5*sin(c + d*x)**4/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2
205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*
a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 30*a**5*sin
(c + d*x)**2*cos(c + d*x)**2/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**
2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 7
35*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 15*a**5*cos(c + d*x)**4/(105*a**12*b*d + 7
35*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**
8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*
d*sin(c + d*x)**7) - 49*a**4*b*sin(c + d*x)**5/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**
3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*s
in(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 70*a**4*b*sin(c + d*x)**
3*cos(c + d*x)**2/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a*
*9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**
7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 63*a**3*b**2*sin(c + d*x)**6/(105*a**12*b*d + 735*a**
11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5
*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(
c + d*x)**7) - 70*a**3*b**2*sin(c + d*x)**4*cos(c + d*x)**2/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2
205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*
a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 41*a**2*b**
3*sin(c + d*x)**7/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a*
*9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**
7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 42*a**2*b**3*sin(c + d*x)**5*cos(c + d*x)**2/(105*a**
12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3
+ 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*
a**5*b**8*d*sin(c + d*x)**7) - 14*a*b**4*sin(c + d*x)**8/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205
*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**
7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 14*a*b**4*sin(
c + d*x)**6*cos(c + d*x)**2/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2
+ 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 73
5*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7) - 2*b**5*sin(c + d*x)**9/(105*a**12*b*d + 735
*a**11*b**2*d*sin(c + d*x) + 2205*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*
b**5*d*sin(c + d*x)**4 + 2205*a**7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*
sin(c + d*x)**7) - 2*b**5*sin(c + d*x)**7*cos(c + d*x)**2/(105*a**12*b*d + 735*a**11*b**2*d*sin(c + d*x) + 220
5*a**10*b**3*d*sin(c + d*x)**2 + 3675*a**9*b**4*d*sin(c + d*x)**3 + 3675*a**8*b**5*d*sin(c + d*x)**4 + 2205*a*
*7*b**6*d*sin(c + d*x)**5 + 735*a**6*b**7*d*sin(c + d*x)**6 + 105*a**5*b**8*d*sin(c + d*x)**7), True))

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Giac [A]  time = 1.44224, size = 158, normalized size = 1.12 \begin{align*} -\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} \end{align*}

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)