\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [1183]

Optimal result

Integrand size = 31, antiderivative size = 469 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac {8 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^7 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-2/3*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))^(3/2)+2/3*(13*a^2-5*b^2)*cos(d*x+c)*sin(d*x+c)
^4/a^2/b^2/d/(a+b*sin(d*x+c))^(1/2)+128/315*a*(40*a^2-19*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^6/d-8/315*(4
80*a^2-203*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+4/63*(160*a^2-63*b^2)*cos(d*x+c)*sin(d*x+c)
^2*(a+b*sin(d*x+c))^(1/2)/a/b^4/d-10/9*(8*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/a^2/b^3/d-
8/315*(1280*a^4-768*a^2*b^2+21*b^4)*(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(co
s(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^7/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+8
/315*a*(1280*a^4-1088*a^2*b^2+123*b^4)*(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^7/d/(a+b*sin(d*x+c))^(1/2
)

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2970, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac {8 a \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{315 b^7 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(13*a^2 - 5*b^2)*Cos[
c + d*x]*Sin[c + d*x]^4)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (128*a*(40*a^2 - 19*b^2)*Cos[c + d*x]*Sqrt[a
+ b*Sin[c + d*x]])/(315*b^6*d) - (8*(480*a^2 - 203*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(
315*b^5*d) + (4*(160*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b^4*d) - (10*(8
*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*a^2*b^3*d) + (8*(1280*a^4 - 768*a^2*b^2
+ 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^7*d*Sqrt[(a + b*Sin[c
+ d*x])/(a + b)]) - (8*a*(1280*a^4 - 1088*a^2*b^2 + 123*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqr
t[(a + b*Sin[c + d*x])/(a + b)])/(315*b^7*d*Sqrt[a + b*Sin[c + d*x]])

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 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 2970

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f
*(m + 1))), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b
*Sin[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2))), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n +
4, 0])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \int \frac {\sin ^3(c+d x) \left (\frac {1}{4} \left (96 a^2-35 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {15}{4} \left (8 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 a^2 b^2} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac {8 \int \frac {\sin ^2(c+d x) \left (-\frac {45}{4} a \left (8 a^2-3 b^2\right )+3 a^2 b \sin (c+d x)+\frac {3}{4} a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{27 a^2 b^3} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac {16 \int \frac {\sin (c+d x) \left (\frac {3}{2} a^2 \left (160 a^2-63 b^2\right )-15 a^3 b \sin (c+d x)-\frac {3}{4} a^2 \left (480 a^2-203 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{189 a^2 b^4} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac {32 \int \frac {-\frac {3}{4} a^3 \left (480 a^2-203 b^2\right )+\frac {3}{8} a^2 b \left (160 a^2-21 b^2\right ) \sin (c+d x)+18 a^3 \left (40 a^2-19 b^2\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{945 a^2 b^5} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac {64 \int \frac {-\frac {9}{8} a^3 b \left (160 a^2-51 b^2\right )-\frac {9}{16} a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{2835 a^2 b^6} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac {\left (4 \left (1280 a^4-768 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{315 b^7}-\frac {\left (4 a \left (1280 a^4-1088 a^2 b^2+123 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 b^7} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac {\left (4 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{315 b^7 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a \left (1280 a^4-1088 a^2 b^2+123 b^4\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}{315 b^7 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac {8 \left (1280 a^4-768 a^2 b^2+21 b^4\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)}}{315 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (1280 a^4-1088 a^2 b^2+123 b^4\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}}}{315 b^7 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1044\) vs. \(2(469)=938\).

Time = 7.54 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.23 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {315 \left (\frac {\left (\left (a^2+3 b^2\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+a (-a+b) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2 b}-\frac {\cos (c+d x) \left (2 a \left (a^2+b^2\right )+b \left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}\right )+\frac {315 \left (\frac {\left (\left (32 a^4-57 a^2 b^2+21 b^4\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+a \left (-32 a^3+32 a^2 b+33 a b^2-33 b^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2}-\frac {b \left (4 a \left (8 a^4-13 a^2 b^2+3 b^4\right ) \cos (c+d x)+b \left (20 a^4-33 a^2 b^2+9 b^4\right ) \sin (2 (c+d x))\right )}{2 \left (a^2-b^2\right )^2}\right )}{b^3}-\frac {21 \left (\frac {\left (\left (-2048 a^6+4192 a^4 b^2-2355 a^2 b^4+231 b^6\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+a \left (2048 a^5-2048 a^4 b-2656 a^3 b^2+2656 a^2 b^3+603 a b^4-603 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2}+\frac {b \cos (c+d x) \left (-64 a b^2 \left (a^2-b^2\right )^2 \cos (2 (c+d x))+b \left (1280 a^6-2536 a^4 b^2+1347 a^2 b^4-111 b^6\right ) \sin (c+d x)+2 \left (512 a^7-952 a^5 b^2+423 a^3 b^4+7 a b^6+6 b^3 \left (a^2-b^2\right )^2 \sin (3 (c+d x))\right )\right )}{\left (a^2-b^2\right )^2}\right )}{b^5}-\frac {5 (a+b \sin (c+d x)) \left (\frac {\left (-4 b \left (-4096 a^7 b+8960 a^5 b^3-5884 a^3 b^5+1041 a b^7\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )+\left (65536 a^8-161792 a^6 b^2+129664 a^4 b^4-35109 a^2 b^6+1617 b^8\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 ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{(a-b)^2 (a+b)^2}+b (a+b \sin (c+d x)) \left (-128 a \left (88 a^2-27 b^2\right ) \cos (c+d x)+416 a b^2 \cos (3 (c+d x))+\frac {21 a \left (64 a^6-112 a^4 b^2+56 a^2 b^4-7 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {21 \left (1088 a^8-2576 a^6 b^2+1960 a^4 b^4-497 a^2 b^6+21 b^8\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-8 b \left (-276 a^2+35 b^2\right ) \sin (2 (c+d x))-56 b^3 \sin (4 (c+d x))\right )\right )}{b^7}}{10080 d (a+b \sin (c+d x))^{3/2}} \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(315*((((a^2 + 3*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(-a + b)*EllipticF[(-2*c + Pi - 2*d*
x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^2*b) - (Cos[c + d*x]*(2*a*(a^2 + b^2) + b
*(a^2 + 3*b^2)*Sin[c + d*x]))/(a^2 - b^2)^2) + (315*((((32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(-2*c + Pi - 2
*d*x)/4, (2*b)/(a + b)] + a*(-32*a^3 + 32*a^2*b + 33*a*b^2 - 33*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a
+ b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/(a - b)^2 - (b*(4*a*(8*a^4 - 13*a^2*b^2 + 3*b^4)*Cos[c + d*x] +
b*(20*a^4 - 33*a^2*b^2 + 9*b^4)*Sin[2*(c + d*x)]))/(2*(a^2 - b^2)^2)))/b^3 - (21*((((-2048*a^6 + 4192*a^4*b^2
- 2355*a^2*b^4 + 231*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(2048*a^5 - 2048*a^4*b - 2656*a^
3*b^2 + 2656*a^2*b^3 + 603*a*b^4 - 603*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d
*x])/(a + b))^(3/2))/(a - b)^2 + (b*Cos[c + d*x]*(-64*a*b^2*(a^2 - b^2)^2*Cos[2*(c + d*x)] + b*(1280*a^6 - 253
6*a^4*b^2 + 1347*a^2*b^4 - 111*b^6)*Sin[c + d*x] + 2*(512*a^7 - 952*a^5*b^2 + 423*a^3*b^4 + 7*a*b^6 + 6*b^3*(a
^2 - b^2)^2*Sin[3*(c + d*x)])))/(a^2 - b^2)^2))/b^5 - (5*(a + b*Sin[c + d*x])*(((-4*b*(-4096*a^7*b + 8960*a^5*
b^3 - 5884*a^3*b^5 + 1041*a*b^7)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (65536*a^8 - 161792*a^6*b^2
+ 129664*a^4*b^4 - 35109*a^2*b^6 + 1617*b^8)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*Ell
ipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/((a - b)^2*(a + b)^2) + b*(
a + b*Sin[c + d*x])*(-128*a*(88*a^2 - 27*b^2)*Cos[c + d*x] + 416*a*b^2*Cos[3*(c + d*x)] + (21*a*(64*a^6 - 112*
a^4*b^2 + 56*a^2*b^4 - 7*b^6)*Cos[c + d*x])/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) - (21*(1088*a^8 - 2576*a^6*b^
2 + 1960*a^4*b^4 - 497*a^2*b^6 + 21*b^8)*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])) - 8*b*(-276*a^2 +
35*b^2)*Sin[2*(c + d*x)] - 56*b^3*Sin[4*(c + d*x)])))/b^7)/(10080*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. \(2032\) vs. \(2(499)=998\).

Time = 2.61 (sec) , antiderivative size = 2033, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(2033\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/315*(-35*b^7*cos(d*x+c)^6*sin(d*x+c)+60*a*b^6*cos(d*x+c)^6+(120*a^2*b^5-7*b^7)*cos(d*x+c)^4*sin(d*x+c)+(-320
*a^3*b^4+102*a*b^6)*cos(d*x+c)^4+(3200*a^4*b^3-1740*a^2*b^5+42*b^7)*cos(d*x+c)^2*sin(d*x+c)+(2560*a^5*b^2-896*
a^3*b^4-162*a*b^6)*cos(d*x+c)^2+4*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(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*(1280*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b
-960*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2-1088*EllipticF((b/(a-b)*sin(d*x
+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3+666*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))
^(1/2))*a^2*b^4+123*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^5-21*EllipticF((b/(a
-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^6-1280*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b
)/(a+b))^(1/2))*a^6+2048*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2-789*Ellipti
cE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4+21*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(
1/2),((a-b)/(a+b))^(1/2))*b^6)*sin(d*x+c)+5120*(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-3840*(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^5*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)*EllipticF((b/(a-b)*sin(d*x
+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3+2664*(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^4+492*(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^5-84*(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)*si
n(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6-5120*(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^7+8192*(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^5*b^2-3156*(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^3*b^4+84*(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*b^6)/(a+b*sin(d*x+c))^(3/2)/b^8/cos(d*x+c)/d

Fricas [C] (verification not implemented)

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

Time = 0.36 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.04 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/945*(8*(sqrt(2)*(640*a^5*b^2 - 624*a^3*b^4 + 87*a*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(640*a^6*b - 624*a^4*b^3
+ 87*a^2*b^5)*sin(d*x + c) - sqrt(2)*(640*a^7 + 16*a^5*b^2 - 537*a^3*b^4 + 87*a*b^6))*sqrt(I*b)*weierstrassPIn
verse(-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)*(640*a^5*b^2 - 624*a^3*b^4 + 87*a*b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(640*a^6*b - 624*a^4*
b^3 + 87*a^2*b^5)*sin(d*x + c) - sqrt(2)*(640*a^7 + 16*a^5*b^2 - 537*a^3*b^4 + 87*a*b^6))*sqrt(-I*b)*weierstra
ssPInverse(-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) - 6*(sqrt(2)*(-1280*I*a^4*b^3 + 768*I*a^2*b^5 - 21*I*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(1280*I*a
^5*b^2 - 768*I*a^3*b^4 + 21*I*a*b^6)*sin(d*x + c) + sqrt(2)*(1280*I*a^6*b + 512*I*a^4*b^3 - 747*I*a^2*b^5 + 21
*I*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, weierstrassPInve
rse(-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)) - 6*(sqrt(2)*(1280*I*a^4*b^3 - 768*I*a^2*b^5 + 21*I*b^7)*cos(d*x + c)^2 + 2*sqrt(2)*(-1280*I*a^5*b^2
+ 768*I*a^3*b^4 - 21*I*a*b^6)*sin(d*x + c) + sqrt(2)*(-1280*I*a^6*b - 512*I*a^4*b^3 + 747*I*a^2*b^5 - 21*I*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*(60*a*b^6*cos(d*x + c)^5 - 2*(160*a^3*b^4 - 51*a*b^6)*cos(d*x + c)^3 + 2*(1280*a^5*b^2 - 448*a^3*b^4
- 81*a*b^6)*cos(d*x + c) - (35*b^7*cos(d*x + c)^5 - (120*a^2*b^5 - 7*b^7)*cos(d*x + c)^3 - 2*(1600*a^4*b^3 -
870*a^2*b^5 + 21*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^10*d*cos(d*x + c)^2 - 2*a*b^9*d
*sin(d*x + c) - (a^2*b^8 + b^10)*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^3/(b*sin(d*x + c) + a)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2),x)

[Out]

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2), x)