\(\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) [1151]

Optimal result

Integrand size = 31, antiderivative size = 528 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{45045 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

16/45045*a*(32*a^4-47*a^2*b^2-27*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/b^5/d-8/45045*(160*a^4-375*a^2*b^2+117
*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(5/2)/b^5/d+8/1287*a*(8*a^2-21*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(
5/2)/b^4/d-2/2145*(80*a^2-221*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(5/2)/b^3/d+4/39*a*cos(d*x+c)*sin(
d*x+c)^3*(a+b*sin(d*x+c))^(5/2)/b^2/d-2/15*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(5/2)/b/d+8/45045*(64*a^6-
174*a^4*b^2+81*a^2*b^4-195*b^6)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+16/45045*a*(32*a^6-111*a^4*b^2+102*a^2
*b^4-471*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)*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^6/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-8/45045*(64*a^8-238*a^6
*b^2+255*a^4*b^4-276*a^2*b^6+195*b^8)*(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^6/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2974, 3128, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {8 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{45045 b^6 d \sqrt {a+b \sin (c+d x)}}+\frac {4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d} \]

[In]

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

[Out]

(8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) + (16*a*
(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*
b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin
[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*
Sin[c + d*x])^(5/2))/(2145*b^3*d) + (4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) -
(2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^
4 - 471*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(a + b*S
in[c + d*x])/(a + b)]) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*EllipticF[(c - Pi/2 +
d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*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 2832

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

Rule 2974

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m
+ n + 3)*(m + n + 4))), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x
])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*
(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e
+ f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[
m, 0] || IntegersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[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 {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {4 \int \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {15}{4} \left (4 a^2-13 b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-221 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{195 b^2} \\ & = -\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {8 \int \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac {1}{2} a \left (80 a^2-221 b^2\right )-\frac {3}{2} b \left (5 a^2+13 b^2\right ) \sin (c+d x)+\frac {15}{2} a \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2145 b^3} \\ & = \frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 \int (a+b \sin (c+d x))^{3/2} \left (\frac {15}{2} a^2 \left (8 a^2-21 b^2\right )+6 a b \left (5 a^2-9 b^2\right ) \sin (c+d x)-\frac {3}{4} \left (160 a^4-375 a^2 b^2+117 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{19305 b^4} \\ & = -\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {32 \int (a+b \sin (c+d x))^{3/2} \left (-\frac {45}{8} b \left (16 a^4-27 a^2 b^2+39 b^4\right )+\frac {15}{4} a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \sin (c+d x)\right ) \, dx}{135135 b^5} \\ & = \frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {64 \int \sqrt {a+b \sin (c+d x)} \left (-\frac {45}{16} a b \left (16 a^4-41 a^2 b^2+249 b^4\right )+\frac {45}{16} \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \sin (c+d x)\right ) \, dx}{675675 b^5} \\ & = \frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {128 \int \frac {\frac {45}{32} b \left (16 a^6-51 a^4 b^2-666 a^2 b^4-195 b^6\right )+\frac {45}{16} a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{2027025 b^5} \\ & = \frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{45045 b^6}+\frac {\left (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^6} \\ & = \frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{45045 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\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}{45045 b^6 \sqrt {a+b \sin (c+d x)}} \\ & = \frac {8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{45045 b^5 d}+\frac {16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac {8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac {8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac {2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac {4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac {16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\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)}}{45045 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\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}}}{45045 b^6 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.64 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.72 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (512 \left (32 a^7-111 a^5 b^2+102 a^3 b^4-471 a b^6\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-256 \left (64 a^7-64 a^6 b-174 a^5 b^2+174 a^4 b^3+81 a^3 b^4-81 a^2 b^5-195 a b^6+195 b^7\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (4096 a^6-12416 a^4 b^2+8100 a^2 b^4+6786 b^6+\left (-1280 a^4 b^2+3168 a^2 b^4+21723 b^6\right ) \cos (2 (c+d x))+42 \left (6 a^2 b^4-13 b^6\right ) \cos (4 (c+d x))-3003 b^6 \cos (6 (c+d x))-3072 a^5 b \sin (c+d x)+8432 a^3 b^3 \sin (c+d x)-41424 a b^5 \sin (c+d x)+560 a^3 b^3 \sin (3 (c+d x))+13776 a b^5 \sin (3 (c+d x))+7392 a b^5 \sin (5 (c+d x))\right )\right )}{1441440 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]

[In]

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

[Out]

(Sqrt[a + b*Sin[c + d*x]]*(512*(32*a^7 - 111*a^5*b^2 + 102*a^3*b^4 - 471*a*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/
4, (2*b)/(a + b)] - 256*(64*a^7 - 64*a^6*b - 174*a^5*b^2 + 174*a^4*b^3 + 81*a^3*b^4 - 81*a^2*b^5 - 195*a*b^6 +
195*b^7)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 2*b*Cos[c + d*x]*Sqrt[(a + b*Sin[c + d*x])/(a + b)
]*(4096*a^6 - 12416*a^4*b^2 + 8100*a^2*b^4 + 6786*b^6 + (-1280*a^4*b^2 + 3168*a^2*b^4 + 21723*b^6)*Cos[2*(c +
d*x)] + 42*(6*a^2*b^4 - 13*b^6)*Cos[4*(c + d*x)] - 3003*b^6*Cos[6*(c + d*x)] - 3072*a^5*b*Sin[c + d*x] + 8432*
a^3*b^3*Sin[c + d*x] - 41424*a*b^5*Sin[c + d*x] + 560*a^3*b^3*Sin[3*(c + d*x)] + 13776*a*b^5*Sin[3*(c + d*x)]
+ 7392*a*b^5*Sin[5*(c + d*x)])))/(1441440*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1800\) vs. \(2(554)=1108\).

Time = 2.94 (sec) , antiderivative size = 1801, normalized size of antiderivative = 3.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1801\)

[In]

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

[Out]

-2/45045*(780*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*E
llipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^9-256*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x
+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(
1/2))*a^9-428*a^5*b^4+128*a^7*b^2+256*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(
d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^8*b-192*((a+b*sin(d*x+c
))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a
-b))^(1/2),((a-b)/(a+b))^(1/2))*a^7*b^2-952*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(
1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b^3+684*((a+b*s
in(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*
x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^4+1020*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^
(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^5-34
80*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((
a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^6-1104*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)
*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*
a^2*b^7+2988*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*El
lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^8+1144*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d
*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))
^(1/2))*a^7*b^2-1704*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^
(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^4+4584*((a+b*sin(d*x+c))/(a-b))^(1/2
)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a
-b)/(a+b))^(1/2))*a^3*b^6-3768*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))
*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^8+780*a*b^8+360*a^3*b^6-6699
*a*b^8*sin(d*x+c)^8-3759*a^2*b^7*sin(d*x+c)^7+7*a^3*b^6*sin(d*x+c)^6+17682*a*b^8*sin(d*x+c)^6-10*a^4*b^5*sin(d
*x+c)^5+10362*a^2*b^7*sin(d*x+c)^5+16*a^5*b^4*sin(d*x+c)^4-62*a^3*b^6*sin(d*x+c)^4-12603*a*b^8*sin(d*x+c)^4-32
*a^6*b^3*sin(d*x+c)^3+122*a^4*b^5*sin(d*x+c)^3-8115*a^2*b^7*sin(d*x+c)^3-128*a^7*b^2*sin(d*x+c)^2+412*a^5*b^4*
sin(d*x+c)^2-305*a^3*b^6*sin(d*x+c)^2+840*a*b^8*sin(d*x+c)^2+32*a^6*b^3*sin(d*x+c)-112*a^4*b^5*sin(d*x+c)+1512
*a^2*b^7*sin(d*x+c)-3003*b^9*sin(d*x+c)^9+7644*b^9*sin(d*x+c)^7-5109*b^9*sin(d*x+c)^5-312*b^9*sin(d*x+c)^3+780
*b^9*sin(d*x+c))/b^7/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.22 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.30 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (128 \, a^{8} - 492 \, a^{6} b^{2} + 561 \, a^{4} b^{4} + 114 \, a^{2} b^{6} + 585 \, b^{8}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (128 \, a^{8} - 492 \, a^{6} b^{2} + 561 \, a^{4} b^{4} + 114 \, a^{2} b^{6} + 585 \, b^{8}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 12 \, \sqrt {2} {\left (-32 i \, a^{7} b + 111 i \, a^{5} b^{3} - 102 i \, a^{3} b^{5} + 471 i \, a b^{7}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 12 \, \sqrt {2} {\left (32 i \, a^{7} b - 111 i \, a^{5} b^{3} + 102 i \, a^{3} b^{5} - 471 i \, a b^{7}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (3003 \, b^{8} \cos \left (d x + c\right )^{7} - 21 \, {\left (3 \, a^{2} b^{6} + 208 \, b^{8}\right )} \cos \left (d x + c\right )^{5} + 5 \, {\left (16 \, a^{4} b^{4} - 27 \, a^{2} b^{6} + 39 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (64 \, a^{6} b^{2} - 174 \, a^{4} b^{4} + 81 \, a^{2} b^{6} - 195 \, b^{8}\right )} \cos \left (d x + c\right ) - 2 \, {\left (1848 \, a b^{7} \cos \left (d x + c\right )^{5} + 35 \, {\left (a^{3} b^{5} - 15 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (16 \, a^{5} b^{3} - 41 \, a^{3} b^{5} + 249 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{135135 \, b^{7} d} \]

[In]

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

[Out]

2/135135*(2*sqrt(2)*(128*a^8 - 492*a^6*b^2 + 561*a^4*b^4 + 114*a^2*b^6 + 585*b^8)*sqrt(I*b)*weierstrassPInvers
e(-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) + 2*sqrt(2)*(128*a^8 - 492*a^6*b^2 + 561*a^4*b^4 + 114*a^2*b^6 + 585*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) - 12*sqrt(2)*(-32*I*a^7*b + 111*I*a^5*b^3 - 102*I*a^3*b^5 + 471*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)) - 12*sqrt(2)*(32*I*a^7*b - 111*
I*a^5*b^3 + 102*I*a^3*b^5 - 471*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*(3003*b^8*cos(d*x + c)^7 - 21*(3*a^2*b^6 + 208*b^8)*cos(d*x +
c)^5 + 5*(16*a^4*b^4 - 27*a^2*b^6 + 39*b^8)*cos(d*x + c)^3 - 2*(64*a^6*b^2 - 174*a^4*b^4 + 81*a^2*b^6 - 195*b^
8)*cos(d*x + c) - 2*(1848*a*b^7*cos(d*x + c)^5 + 35*(a^3*b^5 - 15*a*b^7)*cos(d*x + c)^3 - 3*(16*a^5*b^3 - 41*a
^3*b^5 + 249*a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^7*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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