\(\int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [463]

Optimal result

Integrand size = 21, antiderivative size = 77 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac {a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac {1}{5 b^3 d (a+b \sin (c+d x))^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac {a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac {1}{5 b^3 d (a+b \sin (c+d x))^5} \]

[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 711

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 2747

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b^2-x^2}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = \frac {\text {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] (verified)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a^2-15 b^2+7 a b \sin (c+d x)+21 b^2 \sin ^2(c+d x)}{105 b^3 d (a+b \sin (c+d x))^7} \]

[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)

Maple [A] (verified)

Time = 7.59 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {1}{5 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {a}{3 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {-a^{2}+b^{2}}{7 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{7}}}{d}\)	\(67\)
default	\(\frac {\frac {1}{5 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {a}{3 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {-a^{2}+b^{2}}{7 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{7}}}{d}\)	\(67\)
risch	\(\frac {32 i \left (14 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+21 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-14 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-4 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}+18 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+21 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}\right )}{105 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{7} d \,b^{3}}\)	\(125\)
parallelrisch	\(\frac {\left (560 a^{6}-7980 a^{4} b^{2}-6279 a^{2} b^{4}-315 b^{6}\right ) \sin \left (3 d x +3 c \right )+\left (-8960 a^{5} b -16590 a^{3} b^{3}-3150 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-280 a^{5} b +4116 a^{3} b^{3}+1260 a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+\left (-84 a^{4} b^{2}+1253 a^{2} b^{4}+105 b^{6}\right ) \sin \left (5 d x +5 c \right )+\left (14 a^{3} b^{3}-210 a \,b^{5}\right ) \cos \left (6 d x +6 c \right )+\left (a^{2} b^{4}-15 b^{6}\right ) \sin \left (7 d x +7 c \right )+\left (5040 a^{6}+24360 a^{4} b^{2}+12565 a^{2} b^{4}+525 b^{6}\right ) \sin \left (d x +c \right )+9240 a^{5} b +12460 a^{3} b^{3}+2100 a \,b^{5}}{6720 a^{7} d \left (a +b \sin \left (d x +c \right )\right )^{7}}\)	\(255\)

[In]

int(cos(d*x+c)^3/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (71) = 142\).

Time = 0.34 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.30 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {21 \, b^{2} \cos \left (d x + c\right )^{2} - 7 \, a b \sin \left (d x + c\right ) - a^{2} - 6 \, b^{2}}{105 \, {\left (7 \, a b^{9} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{5} + 10 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{3} + 21 \, a^{5} b^{5} + 35 \, a^{3} b^{7} + 7 \, a b^{9}\right )} d + {\left (b^{10} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{6} + 42 \, a^{2} b^{8} + 3 \, b^{10}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{4} + 35 \, a^{4} b^{6} + 21 \, a^{2} b^{8} + b^{10}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]

[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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 636 vs. \(2 (66) = 132\).

Time = 9.61 (sec) , antiderivative size = 636, normalized size of antiderivative = 8.26 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\begin {cases} \frac {x \cos ^{3}{\left (c \right )}}{a^{8}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\frac {2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {\sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}}{a^{8}} & \text {for}\: b = 0 \\\frac {x \cos ^{3}{\left (c \right )}}{\left (a + b \sin {\left (c \right )}\right )^{8}} & \text {for}\: d = 0 \\\frac {a^{2}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} + \frac {7 a b \sin {\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} + \frac {6 b^{2} \sin ^{2}{\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} - \frac {15 b^{2} \cos ^{2}{\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**3/(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((x*cos(c)**3/a**8, Eq(b, 0) & Eq(d, 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)), (a**2/(105*a**7*b**3*d + 735*a**6*b**4*d*sin(c +
d*x) + 2205*a**5*b**5*d*sin(c + d*x)**2 + 3675*a**4*b**6*d*sin(c + d*x)**3 + 3675*a**3*b**7*d*sin(c + d*x)**4
+ 2205*a**2*b**8*d*sin(c + d*x)**5 + 735*a*b**9*d*sin(c + d*x)**6 + 105*b**10*d*sin(c + d*x)**7) + 7*a*b*sin(c
+ d*x)/(105*a**7*b**3*d + 735*a**6*b**4*d*sin(c + d*x) + 2205*a**5*b**5*d*sin(c + d*x)**2 + 3675*a**4*b**6*d*
sin(c + d*x)**3 + 3675*a**3*b**7*d*sin(c + d*x)**4 + 2205*a**2*b**8*d*sin(c + d*x)**5 + 735*a*b**9*d*sin(c + d
*x)**6 + 105*b**10*d*sin(c + d*x)**7) + 6*b**2*sin(c + d*x)**2/(105*a**7*b**3*d + 735*a**6*b**4*d*sin(c + d*x)
+ 2205*a**5*b**5*d*sin(c + d*x)**2 + 3675*a**4*b**6*d*sin(c + d*x)**3 + 3675*a**3*b**7*d*sin(c + d*x)**4 + 22
05*a**2*b**8*d*sin(c + d*x)**5 + 735*a*b**9*d*sin(c + d*x)**6 + 105*b**10*d*sin(c + d*x)**7) - 15*b**2*cos(c +
d*x)**2/(105*a**7*b**3*d + 735*a**6*b**4*d*sin(c + d*x) + 2205*a**5*b**5*d*sin(c + d*x)**2 + 3675*a**4*b**6*d
*sin(c + d*x)**3 + 3675*a**3*b**7*d*sin(c + d*x)**4 + 2205*a**2*b**8*d*sin(c + d*x)**5 + 735*a*b**9*d*sin(c +
d*x)**6 + 105*b**10*d*sin(c + d*x)**7), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (71) = 142\).

Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.96 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \, {\left (b^{10} \sin \left (d x + c\right )^{7} + 7 \, a b^{9} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{8} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{7} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{6} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{5} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{4} \sin \left (d x + c\right ) + a^{7} b^{3}\right )} d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{3} d} \]

[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)

Mupad [B] (verification not implemented)

Time = 4.74 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.97 \[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\frac {a^2-15\,b^2}{105\,b^3}+\frac {{\sin \left (c+d\,x\right )}^2}{5\,b}+\frac {a\,\sin \left (c+d\,x\right )}{15\,b^2}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]

[In]

int(cos(c + d*x)^3/(a + b*sin(c + d*x))^8,x)

[Out]

((a^2 - 15*b^2)/(105*b^3) + sin(c + d*x)^2/(5*b) + (a*sin(c + d*x))/(15*b^2))/(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)))