3.461 \(\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)
Optimal. Leaf size=207 \[ \frac{\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac{3 \left (-6 a^2 b^2+5 a^4+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}+\frac{1}{b^7 d (a+b \sin (c+d x))}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2} \]
[Out]
(a^2 - b^2)^3/(7*b^7*d*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x])^6) + (3*(5*a^4
- 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*Sin[c + d*x])^4) + (5
*a^2 - b^2)/(b^7*d*(a + b*Sin[c + d*x])^3) - (3*a)/(b^7*d*(a + b*Sin[c + d*x])^2) + 1/(b^7*d*(a + b*Sin[c + d*
x]))
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Rubi [A] time = 0.171168, antiderivative size = 207, 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} \[ \frac{\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac{3 \left (-6 a^2 b^2+5 a^4+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}+\frac{1}{b^7 d (a+b \sin (c+d x))}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]
[Out]
(a^2 - b^2)^3/(7*b^7*d*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x])^6) + (3*(5*a^4
- 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*Sin[c + d*x])^4) + (5
*a^2 - b^2)/(b^7*d*(a + b*Sin[c + d*x])^3) - (3*a)/(b^7*d*(a + b*Sin[c + d*x])^2) + 1/(b^7*d*(a + b*Sin[c + d*
x]))
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 ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^3}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{\left (a^2-b^2\right )^3}{(a+x)^8}+\frac{6 a \left (a^2-b^2\right )^2}{(a+x)^7}-\frac{3 \left (5 a^4-6 a^2 b^2+b^4\right )}{(a+x)^6}+\frac{4 \left (5 a^3-3 a b^2\right )}{(a+x)^5}-\frac{3 \left (5 a^2-b^2\right )}{(a+x)^4}+\frac{6 a}{(a+x)^3}-\frac{1}{(a+x)^2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}-\frac{a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2}+\frac{1}{b^7 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.02218, size = 171, normalized size = 0.83 \[ \frac{\frac{\left (a^2-b^2\right )^3}{7 (a+b \sin (c+d x))^7}-\frac{a \left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{(a+b \sin (c+d x))^3}-\frac{a \left (5 a^2-3 b^2\right )}{(a+b \sin (c+d x))^4}+\frac{3 \left (-6 a^2 b^2+5 a^4+b^4\right )}{5 (a+b \sin (c+d x))^5}+\frac{1}{a+b \sin (c+d x)}-\frac{3 a}{(a+b \sin (c+d x))^2}}{b^7 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]
[Out]
((a^2 - b^2)^3/(7*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(a + b*Sin[c + d*x])^6 + (3*(5*a^4 - 6*a^2*b^2 +
b^4))/(5*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(a + b*Sin[c + d*x])^4 + (5*a^2 - b^2)/(a + b*Sin[c +
d*x])^3 - (3*a)/(a + b*Sin[c + d*x])^2 + (a + b*Sin[c + d*x])^(-1))/(b^7*d)
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Maple [A] time = 0.154, size = 208, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{-{a}^{6}+3\,{a}^{4}{b}^{2}-3\,{a}^{2}{b}^{4}+{b}^{6}}{7\,{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{a \left ({a}^{4}-2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }{{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{1}{{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-3\,{\frac{a}{{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{-15\,{a}^{2}+3\,{b}^{2}}{3\,{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a \left ( 5\,{a}^{2}-3\,{b}^{2} \right ) }{{b}^{7} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{-15\,{a}^{4}+18\,{a}^{2}{b}^{2}-3\,{b}^{4}}{5\,{b}^{7} \left ( a+b\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)^7/(a+b*sin(d*x+c))^8,x)
[Out]
1/d*(-1/7*(-a^6+3*a^4*b^2-3*a^2*b^4+b^6)/b^7/(a+b*sin(d*x+c))^7-a*(a^4-2*a^2*b^2+b^4)/b^7/(a+b*sin(d*x+c))^6+1
/b^7/(a+b*sin(d*x+c))-3*a/b^7/(a+b*sin(d*x+c))^2-1/3*(-15*a^2+3*b^2)/b^7/(a+b*sin(d*x+c))^3-a*(5*a^2-3*b^2)/b^
7/(a+b*sin(d*x+c))^4-1/5*(-15*a^4+18*a^2*b^2-3*b^4)/b^7/(a+b*sin(d*x+c))^5)
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Maxima [A] time = 1.01252, size = 377, normalized size = 1.82 \begin{align*} \frac{35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6} + 35 \,{\left (5 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 35 \,{\left (5 \, a^{3} b^{3} - a b^{5}\right )} \sin \left (d x + c\right )^{3} + 21 \,{\left (5 \, a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{2} + 7 \,{\left (5 \, a^{5} b - a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )}{35 \,{\left (b^{14} \sin \left (d x + c\right )^{7} + 7 \, a b^{13} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{12} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{11} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{10} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{9} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{8} \sin \left (d x + c\right ) + a^{7} b^{7}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
[Out]
1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 5*a^6 - a^4*b^2 + a^2*b^4 - 5*b^6 + 35*(5*a^2*b^4 - b
^6)*sin(d*x + c)^4 + 35*(5*a^3*b^3 - a*b^5)*sin(d*x + c)^3 + 21*(5*a^4*b^2 - a^2*b^4 + b^6)*sin(d*x + c)^2 + 7
*(5*a^5*b - a^3*b^3 + a*b^5)*sin(d*x + c))/((b^14*sin(d*x + c)^7 + 7*a*b^13*sin(d*x + c)^6 + 21*a^2*b^12*sin(d
*x + c)^5 + 35*a^3*b^11*sin(d*x + c)^4 + 35*a^4*b^10*sin(d*x + c)^3 + 21*a^5*b^9*sin(d*x + c)^2 + 7*a^6*b^8*si
n(d*x + c) + a^7*b^7)*d)
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Fricas [A] time = 4.18101, size = 882, normalized size = 4.26 \begin{align*} \frac{35 \, b^{6} \cos \left (d x + c\right )^{6} - 5 \, a^{6} - 104 \, a^{4} b^{2} - 155 \, a^{2} b^{4} - 16 \, b^{6} - 35 \,{\left (5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 7 \,{\left (15 \, a^{4} b^{2} + 47 \, a^{2} b^{4} + 8 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 7 \,{\left (15 \, a b^{5} \cos \left (d x + c\right )^{4} + 5 \, a^{5} b + 24 \, a^{3} b^{3} + 11 \, a b^{5} - 25 \,{\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \,{\left (7 \, a b^{13} d \cos \left (d x + c\right )^{6} - 7 \,{\left (5 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{4} + 7 \,{\left (3 \, a^{5} b^{9} + 10 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{7} b^{7} + 21 \, a^{5} b^{9} + 35 \, a^{3} b^{11} + 7 \, a b^{13}\right )} d +{\left (b^{14} d \cos \left (d x + c\right )^{6} - 3 \,{\left (7 \, a^{2} b^{12} + b^{14}\right )} d \cos \left (d x + c\right )^{4} +{\left (35 \, a^{4} b^{10} + 42 \, a^{2} b^{12} + 3 \, b^{14}\right )} d \cos \left (d x + c\right )^{2} -{\left (7 \, a^{6} b^{8} + 35 \, a^{4} b^{10} + 21 \, a^{2} b^{12} + b^{14}\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)^7/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
[Out]
1/35*(35*b^6*cos(d*x + c)^6 - 5*a^6 - 104*a^4*b^2 - 155*a^2*b^4 - 16*b^6 - 35*(5*a^2*b^4 + 2*b^6)*cos(d*x + c)
^4 + 7*(15*a^4*b^2 + 47*a^2*b^4 + 8*b^6)*cos(d*x + c)^2 - 7*(15*a*b^5*cos(d*x + c)^4 + 5*a^5*b + 24*a^3*b^3 +
11*a*b^5 - 25*(a^3*b^3 + a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(7*a*b^13*d*cos(d*x + c)^6 - 7*(5*a^3*b^11 + 3*a
*b^13)*d*cos(d*x + c)^4 + 7*(3*a^5*b^9 + 10*a^3*b^11 + 3*a*b^13)*d*cos(d*x + c)^2 - (a^7*b^7 + 21*a^5*b^9 + 35
*a^3*b^11 + 7*a*b^13)*d + (b^14*d*cos(d*x + c)^6 - 3*(7*a^2*b^12 + b^14)*d*cos(d*x + c)^4 + (35*a^4*b^10 + 42*
a^2*b^12 + 3*b^14)*d*cos(d*x + c)^2 - (7*a^6*b^8 + 35*a^4*b^10 + 21*a^2*b^12 + b^14)*d)*sin(d*x + c))
________________________________________________________________________________________
Sympy [A] time = 69.0541, size = 2718, normalized size = 13.13 \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)**7/(a+b*sin(d*x+c))**8,x)
[Out]
Piecewise((zoo*x*cos(c)**7/sin(c)**8, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((16/(35*d*sin(c + d*x)) - 8*cos(c + d*
x)**2/(35*d*sin(c + d*x)**3) + 6*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - cos(c + d*x)**6/(7*d*sin(c + d*x)**7
))/b**8, Eq(a, 0)), ((16*sin(c + d*x)**7/(35*d) + 8*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*sin(c + d*x)**3*
cos(c + d*x)**4/d + sin(c + d*x)*cos(c + d*x)**6/d)/a**8, Eq(b, 0)), (x*cos(c)**7/(a + b*sin(c))**8, Eq(d, 0))
, (a**5*sin(c + d*x)**4/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 12
25*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4
*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + 2*a**5*sin(c + d*x)**2*cos(c + d*x)**2/(35*a**10*
b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1
225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*
b**10*d*sin(c + d*x)**7) + a**5*cos(c + d*x)**4/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**
5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*si
n(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + 7*a**4*b*sin(c + d*x)**5/
(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d
*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6
+ 35*a**3*b**10*d*sin(c + d*x)**7) + 14*a**4*b*sin(c + d*x)**3*cos(c + d*x)**2/(35*a**10*b**3*d + 245*a**9*b**
4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c
+ d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)*
*7) + 7*a**4*b*sin(c + d*x)*cos(c + d*x)**4/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*
sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c
+ d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + 16*a**3*b**2*sin(c + d*x)**6/
(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d
*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6
+ 35*a**3*b**10*d*sin(c + d*x)**7) + 27*a**3*b**2*sin(c + d*x)**4*cos(c + d*x)**2/(35*a**10*b**3*d + 245*a**9*
b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*si
n(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*
x)**7) + 6*a**3*b**2*sin(c + d*x)**2*cos(c + d*x)**4/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**
8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8
*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) - 5*a**3*b**2*cos(c +
d*x)**6/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*s
in(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c +
d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + 16*a**2*b**3*sin(c + d*x)**7/(35*a**10*b**3*d + 245*a**9*b**4*d*s
in(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*
x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) +
21*a**2*b**3*sin(c + d*x)**5*cos(c + d*x)**2/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*
d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(
c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + 7*a*b**4*sin(c + d*x)**8/(3
5*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x
)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 +
35*a**3*b**10*d*sin(c + d*x)**7) + 7*a*b**4*sin(c + d*x)**6*cos(c + d*x)**2/(35*a**10*b**3*d + 245*a**9*b**4*d
*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c +
d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7)
+ b**5*sin(c + d*x)**9/(35*a**10*b**3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 12
25*a**7*b**6*d*sin(c + d*x)**3 + 1225*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4
*b**9*d*sin(c + d*x)**6 + 35*a**3*b**10*d*sin(c + d*x)**7) + b**5*sin(c + d*x)**7*cos(c + d*x)**2/(35*a**10*b*
*3*d + 245*a**9*b**4*d*sin(c + d*x) + 735*a**8*b**5*d*sin(c + d*x)**2 + 1225*a**7*b**6*d*sin(c + d*x)**3 + 122
5*a**6*b**7*d*sin(c + d*x)**4 + 735*a**5*b**8*d*sin(c + d*x)**5 + 245*a**4*b**9*d*sin(c + d*x)**6 + 35*a**3*b*
*10*d*sin(c + d*x)**7), True))
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Giac [A] time = 1.44447, size = 290, normalized size = 1.4 \begin{align*} \frac{35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 175 \, a^{2} b^{4} \sin \left (d x + c\right )^{4} - 35 \, b^{6} \sin \left (d x + c\right )^{4} + 175 \, a^{3} b^{3} \sin \left (d x + c\right )^{3} - 35 \, a b^{5} \sin \left (d x + c\right )^{3} + 105 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 21 \, a^{2} b^{4} \sin \left (d x + c\right )^{2} + 21 \, b^{6} \sin \left (d x + c\right )^{2} + 35 \, a^{5} b \sin \left (d x + c\right ) - 7 \, a^{3} b^{3} \sin \left (d x + c\right ) + 7 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6}}{35 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7/(a+b*sin(d*x+c))^8,x, algorithm="giac")
[Out]
1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 175*a^2*b^4*sin(d*x + c)^4 - 35*b^6*sin(d*x + c)^4 +
175*a^3*b^3*sin(d*x + c)^3 - 35*a*b^5*sin(d*x + c)^3 + 105*a^4*b^2*sin(d*x + c)^2 - 21*a^2*b^4*sin(d*x + c)^2
+ 21*b^6*sin(d*x + c)^2 + 35*a^5*b*sin(d*x + c) - 7*a^3*b^3*sin(d*x + c) + 7*a*b^5*sin(d*x + c) + 5*a^6 - a^4*
b^2 + a^2*b^4 - 5*b^6)/((b*sin(d*x + c) + a)^7*b^7*d)