\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1177]

Optimal result

Integrand size = 31, antiderivative size = 401 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {8 \left (320 a^4-318 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 a \left (160 a^4-199 a^2 b^2+39 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^3/a/b^2/d/(a+b*sin(d*x+c))^(1/2)+8/315*a*(160*a^2-139*b^2)*cos(d*x+c)*(a+b*
sin(d*x+c))^(1/2)/b^5/d-16/315*(60*a^2-49*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^4/d+2/63*(80*a^2
-63*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/a/b^3/d-2/9*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(
1/2)/b^2/d-8/315*(320*a^4-318*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)*El
lipticE(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)+16/315*a*(160*a^4-199*a^2*b^2+39*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)*E
llipticF(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 = 0.65 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2971, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^3 d}-\frac {16 a \left (160 a^4-199 a^2 b^2+39 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{315 b^6 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d} \]

[In]

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

[Out]

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (8*a*(160*a^2 - 139*b^2)*Cos
[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d) - (16*(60*a^2 - 49*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*S
in[c + d*x]])/(315*b^4*d) + (2*(80*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b
^3*d) - (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*b^2*d) + (8*(320*a^4 - 318*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^6*d*Sqrt[(a + b*Sin[c + d*x])
/(a + b)]) - (16*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*S
in[c + d*x])/(a + b)])/(315*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 2971

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*b^2*(m + 1)*(m + n + 4)), Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Sim
p[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m
+ n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*((d*Sin[e +
f*x])^(n + 1)/(b^2*d*f*(m + n + 4))), 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] && 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 {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {4 \int \frac {\sin ^2(c+d x) \left (\frac {15}{4} \left (4 a^2-3 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-63 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9 a b^2} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {8 \int \frac {\sin (c+d x) \left (-\frac {1}{2} a \left (80 a^2-63 b^2\right )+\frac {5}{2} a^2 b \sin (c+d x)+a \left (60 a^2-49 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{63 a b^3} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {16 \int \frac {a^2 \left (60 a^2-49 b^2\right )-\frac {1}{4} a b \left (40 a^2-21 b^2\right ) \sin (c+d x)-\frac {3}{4} a^2 \left (160 a^2-139 b^2\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 a b^4} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {32 \int \frac {\frac {3}{8} a^2 b \left (80 a^2-57 b^2\right )+\frac {3}{8} a \left (320 a^4-318 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{945 a b^5} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {\left (4 \left (320 a^4-318 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{315 b^6}-\frac {\left (8 a \left (160 a^4-199 a^2 b^2+39 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 b^6} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {\left (4 \left (320 a^4-318 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^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a \left (160 a^4-199 a^2 b^2+39 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^6 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (160 a^2-139 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}-\frac {16 \left (60 a^2-49 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^4 d}+\frac {2 \left (80 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^3 d}-\frac {2 \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 b^2 d}+\frac {8 \left (320 a^4-318 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^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 a \left (160 a^4-199 a^2 b^2+39 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^6 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.94 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-32 \left (320 a^5+320 a^4 b-318 a^3 b^2-318 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+64 a \left (160 a^4-199 a^2 b^2+39 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (-5120 a^4+4768 a^2 b^2-203 b^4-8 \left (40 a^2 b^2-21 b^4\right ) \cos (2 (c+d x))+35 b^4 \cos (4 (c+d x))-1280 a^3 b \sin (c+d x)+1012 a b^3 \sin (c+d x)+100 a b^3 \sin (3 (c+d x))\right )}{1260 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(-32*(320*a^5 + 320*a^4*b - 318*a^3*b^2 - 318*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2
*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 64*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(-2*c + Pi -
2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(-5120*a^4 + 4768*a^2*b^2 - 203*
b^4 - 8*(40*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 1280*a^3*b*Sin[c + d*x] + 1012*a*b^
3*Sin[c + d*x] + 100*a*b^3*Sin[3*(c + d*x)]))/(1260*b^6*d*Sqrt[a + b*Sin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1189\) vs. \(2(437)=874\).

Time = 2.50 (sec) , antiderivative size = 1190, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1190\)

[In]

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

[Out]

2/315*(35*b^6*sin(d*x+c)^6-50*a*b^5*sin(d*x+c)^5+1280*((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^5*b-960
*((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^2-1592*((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^3+1044*((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)*Elli
pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4+312*((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*b^5-84*((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))*b^6-1280*((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^6+2552*((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^4*b^2-1356*((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^2*b^4+84*((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))*b^6+80*a^2*b^4*sin(d*x+c)^4-112*b^6*s
in(d*x+c)^4-160*a^3*b^3*sin(d*x+c)^3+214*a*b^5*sin(d*x+c)^3-640*a^4*b^2*sin(d*x+c)^2+476*a^2*b^4*sin(d*x+c)^2+
77*b^6*sin(d*x+c)^2+160*a^3*b^3*sin(d*x+c)-164*a*b^5*sin(d*x+c)+640*a^4*b^2-556*a^2*b^4)/b^7/cos(d*x+c)/(a+b*s
in(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 = 723, normalized size of antiderivative = 1.80 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (640 \, a^{5} b - 876 \, a^{3} b^{3} + 213 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (640 \, a^{6} - 876 \, a^{4} b^{2} + 213 \, a^{2} b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (640 \, a^{5} b - 876 \, a^{3} b^{3} + 213 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (640 \, a^{6} - 876 \, a^{4} b^{2} + 213 \, a^{2} b^{4}\right )}\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 ) + 6 \, {\left (\sqrt {2} {\left (320 i \, a^{4} b^{2} - 318 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (320 i \, a^{5} b - 318 i \, a^{3} b^{3} + 21 i \, a b^{5}\right )}\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 ) + 6 \, {\left (\sqrt {2} {\left (-320 i \, a^{4} b^{2} + 318 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-320 i \, a^{5} b + 318 i \, a^{3} b^{3} - 21 i \, a b^{5}\right )}\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 (35 \, b^{6} \cos \left (d x + c\right )^{5} - {\left (80 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (320 \, a^{4} b^{2} - 318 \, a^{2} b^{4} + 21 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (25 \, a b^{5} \cos \left (d x + c\right )^{3} - {\left (80 \, a^{3} b^{3} - 57 \, a b^{5}\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 )}}{945 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]

[In]

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

[Out]

-2/945*(2*(sqrt(2)*(640*a^5*b - 876*a^3*b^3 + 213*a*b^5)*sin(d*x + c) + sqrt(2)*(640*a^6 - 876*a^4*b^2 + 213*a
^2*b^4))*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) + 2*(sqrt(2)*(640*a^5*b - 876*a^3*b^3 + 213*a*b^5)*sin(d*x + c) + s
qrt(2)*(640*a^6 - 876*a^4*b^2 + 213*a^2*b^4))*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) + 6*(sqrt(2)*(320*I*a^4*b^2
- 318*I*a^2*b^4 + 21*I*b^6)*sin(d*x + c) + sqrt(2)*(320*I*a^5*b - 318*I*a^3*b^3 + 21*I*a*b^5))*sqrt(I*b)*weier
strassZeta(-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)) + 6*(sqrt(2)*(-
320*I*a^4*b^2 + 318*I*a^2*b^4 - 21*I*b^6)*sin(d*x + c) + sqrt(2)*(-320*I*a^5*b + 318*I*a^3*b^3 - 21*I*a*b^5))*
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*(35*b^6*cos(d*x + c)^5 - (80*a^2*b^4 - 7*b^6)*cos(d*x + c)^3 - 2*(320*a^4*b^2 - 318*a^2*b^4 + 21*b^6)*c
os(d*x + c) + 2*(25*a*b^5*cos(d*x + c)^3 - (80*a^3*b^3 - 57*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x
+ c) + a))/(b^8*d*sin(d*x + c) + a*b^7*d)

Sympy [F(-1)]

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

[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]

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

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[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)