\(\int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [533]

Optimal result

Integrand size = 23, antiderivative size = 384 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d} \]

[Out]

-2/3*cos(d*x+c)^7/b/d/(a+b*sin(d*x+c))^(3/2)-28/33*cos(d*x+c)^5*(12*a+b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^(1/
2)+40/99*cos(d*x+c)^3*(32*a^2-3*b^2-28*a*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^5/d-16/99*cos(d*x+c)*(128*a^4-
144*a^2*b^2+15*b^4-3*a*b*(32*a^2-31*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^7/d+128/99*a*(8*a^2-9*b^2)*(4*a^
2-3*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(
1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^8/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-32/99*(128*a^6-272*a^4*b^2+1
59*a^2*b^4-15*b^6)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/
2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^8/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2772, 2942, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{99 b^7 d}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}} \]

[In]

Int[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*Cos[c + d*x]^7)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) - (128*a*(8*a^2 - 9*b^2)*(4*a^2 - 3*b^2)*EllipticE[(c -
Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(99*b^8*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*(
128*a^6 - 272*a^4*b^2 + 159*a^2*b^4 - 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c +
d*x])/(a + b)])/(99*b^8*d*Sqrt[a + b*Sin[c + d*x]]) - (28*Cos[c + d*x]^5*(12*a + b*Sin[c + d*x]))/(33*b^3*d*S
qrt[a + b*Sin[c + d*x]]) + (40*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 3*b^2 - 28*a*b*Sin[c + d*x]))
/(99*b^5*d) - (16*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(128*a^4 - 144*a^2*b^2 + 15*b^4 - 3*a*b*(32*a^2 - 31*b
^2)*Sin[c + d*x]))/(99*b^7*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2772

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Dist[g^2*((p - 1)/(b*(m + 1))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2942

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) -
a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + 1)*(m + p + 1
))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2944

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(m + p)*(m + p +
1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {14 \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx}{3 b} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {280 \int \frac {\cos ^4(c+d x) \left (-\frac {b}{2}-6 a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{33 b^3} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}+\frac {160 \int \frac {\cos ^2(c+d x) \left (\frac {3}{4} b \left (4 a^2-3 b^2\right )+\frac {3}{4} a \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^5} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}+\frac {128 \int \frac {-\frac {3}{8} b \left (32 a^4-51 a^2 b^2+15 b^4\right )-\frac {3}{2} a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{297 b^7} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}-\frac {\left (64 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{99 b^8}+\frac {\left (16 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^8} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}-\frac {\left (64 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{99 b^8 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (16 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{99 b^8 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {256 (a+b) \left (b \left (32 a^4 b-51 a^2 b^3+15 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+4 \left (32 a^5-60 a^3 b^2+27 a b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}+\frac {1}{2} b \cos (c+d x) \left (-32768 a^6+55296 a^4 b^2-18144 a^2 b^4-2574 b^6+\left (2048 a^4 b^2-3648 a^2 b^4+1383 b^6\right ) \cos (2 (c+d x))+\left (-96 a^2 b^4+126 b^6\right ) \cos (4 (c+d x))+9 b^6 \cos (6 (c+d x))-40960 a^5 b \sin (c+d x)+74112 a^3 b^3 \sin (c+d x)-30920 a b^5 \sin (c+d x)-384 a^3 b^3 \sin (3 (c+d x))+596 a b^5 \sin (3 (c+d x))+28 a b^5 \sin (5 (c+d x))\right )}{792 b^8 d (a+b \sin (c+d x))^{3/2}} \]

[In]

Integrate[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(256*(a + b)*(b*(32*a^4*b - 51*a^2*b^3 + 15*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + 4*(32*a^5 -
60*a^3*b^2 + 27*a*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*
d*x)/4, (2*b)/(a + b)]))*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (b*Cos[c + d*x]*(-32768*a^6 + 55296*a^4*b^2 -
18144*a^2*b^4 - 2574*b^6 + (2048*a^4*b^2 - 3648*a^2*b^4 + 1383*b^6)*Cos[2*(c + d*x)] + (-96*a^2*b^4 + 126*b^6)
*Cos[4*(c + d*x)] + 9*b^6*Cos[6*(c + d*x)] - 40960*a^5*b*Sin[c + d*x] + 74112*a^3*b^3*Sin[c + d*x] - 30920*a*b
^5*Sin[c + d*x] - 384*a^3*b^3*Sin[3*(c + d*x)] + 596*a*b^5*Sin[3*(c + d*x)] + 28*a*b^5*Sin[5*(c + d*x)]))/2)/(
792*b^8*d*(a + b*Sin[c + d*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2252\) vs. \(2(422)=844\).

Time = 4.44 (sec) , antiderivative size = 2253, normalized size of antiderivative = 5.87

method	result	size
default	\(\text {Expression too large to display}\)	\(2253\)

[In]

int(cos(d*x+c)^8/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/99*(-9*b^8*cos(d*x+c)^8-14*cos(d*x+c)^6*sin(d*x+c)*a*b^7+(24*a^2*b^6-18*b^8)*cos(d*x+c)^6+(48*a^3*b^5-64*a*
b^7)*cos(d*x+c)^4*sin(d*x+c)+(-128*a^4*b^4+204*a^2*b^6-60*b^8)*cos(d*x+c)^4+(1280*a^5*b^3-2328*a^3*b^5+984*a*b
^7)*cos(d*x+c)^2*sin(d*x+c)+(1024*a^6*b^2-1664*a^4*b^4+456*a^2*b^6+120*b^8)*cos(d*x+c)^2-16*(b/(a-b)*sin(d*x+c
)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*b*(128*EllipticE((b/(
a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7-368*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b
)/(a+b))^(1/2))*a^5*b^2+348*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4-108*Elli
pticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6-128*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))
^(1/2),((a-b)/(a+b))^(1/2))*a^6*b+96*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2
+272*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3-189*EllipticF((b/(a-b)*sin(d*x+
c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4-159*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^
(1/2))*a^2*b^5+93*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6+15*EllipticF((b/(a-b
)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^7)*sin(d*x+c)+2048*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(
a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2
),((a-b)/(a+b))^(1/2))*a^7*b-1536*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(
a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b^2-4352*
(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*Ell
ipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^3+3024*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)
*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b)
)^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4+2544*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/
2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b
^5-1488*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(
1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^6-240*(b/(a-b)*sin(d*x+c)+a/(a-b)
)^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+
a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^7-2048*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b)
)^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*
a^8+5888*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^
(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b^2-5568*(b/(a-b)*sin(d*x+c)+a/(a-
b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c
)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4+1728*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(
a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/
2))*a^2*b^6)/(a+b*sin(d*x+c))^(3/2)/b^9/cos(d*x+c)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.28 (sec) , antiderivative size = 1043, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^8/(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/297*(8*(sqrt(2)*(256*a^6*b^2 - 576*a^4*b^4 + 369*a^2*b^6 - 45*b^8)*cos(d*x + c)^2 - 2*sqrt(2)*(256*a^7*b - 5
76*a^5*b^3 + 369*a^3*b^5 - 45*a*b^7)*sin(d*x + c) - sqrt(2)*(256*a^8 - 320*a^6*b^2 - 207*a^4*b^4 + 324*a^2*b^6
- 45*b^8))*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*
cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + 8*(sqrt(2)*(256*a^6*b^2 - 576*a^4*b^4 + 369*a^2*b^6 - 45*b^8)*
cos(d*x + c)^2 - 2*sqrt(2)*(256*a^7*b - 576*a^5*b^3 + 369*a^3*b^5 - 45*a*b^7)*sin(d*x + c) - sqrt(2)*(256*a^8
- 320*a^6*b^2 - 207*a^4*b^4 + 324*a^2*b^6 - 45*b^8))*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2,
-8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 96*(sqrt(2)*(32*I*a
^5*b^3 - 60*I*a^3*b^5 + 27*I*a*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(-32*I*a^6*b^2 + 60*I*a^4*b^4 - 27*I*a^2*b^6)*s
in(d*x + c) + sqrt(2)*(-32*I*a^7*b + 28*I*a^5*b^3 + 33*I*a^3*b^5 - 27*I*a*b^7))*sqrt(I*b)*weierstrassZeta(-4/3
*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*
I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 96*(sqrt(2)*(-32*I*a^5*b^3 +
60*I*a^3*b^5 - 27*I*a*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(32*I*a^6*b^2 - 60*I*a^4*b^4 + 27*I*a^2*b^6)*sin(d*x +
c) + sqrt(2)*(32*I*a^7*b - 28*I*a^5*b^3 - 33*I*a^3*b^5 + 27*I*a*b^7))*sqrt(-I*b)*weierstrassZeta(-4/3*(4*a^2 -
3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 +
9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(9*b^8*cos(d*x + c)^7 - 6*(4*a^2*
b^6 - 3*b^8)*cos(d*x + c)^5 + 4*(32*a^4*b^4 - 51*a^2*b^6 + 15*b^8)*cos(d*x + c)^3 - 8*(128*a^6*b^2 - 208*a^4*b
^4 + 57*a^2*b^6 + 15*b^8)*cos(d*x + c) + 2*(7*a*b^7*cos(d*x + c)^5 - 8*(3*a^3*b^5 - 4*a*b^7)*cos(d*x + c)^3 -
4*(160*a^5*b^3 - 291*a^3*b^5 + 123*a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^11*d*cos(d*
x + c)^2 - 2*a*b^10*d*sin(d*x + c) - (a^2*b^9 + b^11)*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8/(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{8}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^8/(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^8/(b*sin(d*x + c) + a)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^8/(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(cos(c + d*x)^8/(a + b*sin(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)^8/(a + b*sin(c + d*x))^(5/2), x)