\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1176]

Optimal result

Integrand size = 31, antiderivative size = 466 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}-\frac {8 a \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1155 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1155 b^7 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))^(1/2)-8/1155*(640*a^4-592*a^2*b^2+15*b^4)*cos(d*
x+c)*(a+b*sin(d*x+c))^(1/2)/b^6/d+8/1155*a*(480*a^2-419*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/
d-20/231*(32*a^2-27*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/b^4/d+2/33*(40*a^2-33*b^2)*cos(d*x+c)*
sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/a/b^3/d-2/11*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(1/2)/b^2/d+8/1155*a
*(1280*a^4-1344*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)*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^7/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-8/11
55*(1280*a^6-1664*a^4*b^2+369*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)*El
lipticF(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 = 1.05 (sec) , antiderivative size = 466, 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 = {2971, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {8 a \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{1155 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{1155 b^7 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d} \]

[In]

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

[Out]

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(640*a^4 - 592*a^2*b^2 +
15*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^6*d) + (8*a*(480*a^2 - 419*b^2)*Cos[c + d*x]*Sin[c + d*
x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^5*d) - (20*(32*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c
+ d*x]])/(231*b^4*d) + (2*(40*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(33*a*b^3*d
) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b^2*d) - (8*a*(1280*a^4 - 1344*a^2*b^2 + 123*
b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^7*d*Sqrt[(a + b*Sin[c + d*
x])/(a + b)]) + (8*(1280*a^6 - 1664*a^4*b^2 + 369*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)])/(1155*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 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 ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}+\frac {4 \int \frac {\sin ^3(c+d x) \left (\frac {1}{4} \left (96 a^2-77 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {3}{4} \left (40 a^2-33 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{11 a b^2} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}+\frac {8 \int \frac {\sin ^2(c+d x) \left (-\frac {9}{4} a \left (40 a^2-33 b^2\right )+3 a^2 b \sin (c+d x)+\frac {15}{4} a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 a b^3} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}+\frac {16 \int \frac {\sin (c+d x) \left (\frac {15}{2} a^2 \left (32 a^2-27 b^2\right )-\frac {3}{4} a b \left (20 a^2-9 b^2\right ) \sin (c+d x)-\frac {3}{4} a^2 \left (480 a^2-419 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{693 a b^4} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}+\frac {32 \int \frac {-\frac {3}{4} a^3 \left (480 a^2-419 b^2\right )+\frac {3}{8} a^2 b \left (160 a^2-93 b^2\right ) \sin (c+d x)+\frac {9}{8} a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 a b^5} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}+\frac {64 \int \frac {-\frac {9}{16} a b \left (320 a^4-246 a^2 b^2-15 b^4\right )-\frac {9}{16} a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{10395 a b^6} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}-\frac {\left (4 a \left (1280 a^4-1344 a^2 b^2+123 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{1155 b^7}+\frac {\left (4 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{1155 b^7} \\ & = -\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}-\frac {\left (4 a \left (1280 a^4-1344 a^2 b^2+123 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}{1155 b^7 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (1280 a^6-1664 a^4 b^2+369 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}{1155 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)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}-\frac {8 a \left (1280 a^4-1344 a^2 b^2+123 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)}}{1155 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (1280 a^6-1664 a^4 b^2+369 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}}}{1155 b^7 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {64 a \left (1280 a^5+1280 a^4 b-1344 a^3 b^2-1344 a^2 b^3+123 a b^4+123 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-40960 a^5+40448 a^3 b^2-2728 a b^4-16 \left (160 a^3 b^2-93 a b^4\right ) \cos (2 (c+d x))+280 a b^4 \cos (4 (c+d x))-10240 a^4 b \sin (c+d x)+8672 a^2 b^3 \sin (c+d x)+330 b^5 \sin (c+d x)+800 a^2 b^3 \sin (3 (c+d x))-255 b^5 \sin (3 (c+d x))-105 b^5 \sin (5 (c+d x))\right )}{9240 b^7 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(64*a*(1280*a^5 + 1280*a^4*b - 1344*a^3*b^2 - 1344*a^2*b^3 + 123*a*b^4 + 123*b^5)*EllipticE[(-2*c + Pi - 2*d*x
)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*E
llipticF[(-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]*(-40960*a^5
+ 40448*a^3*b^2 - 2728*a*b^4 - 16*(160*a^3*b^2 - 93*a*b^4)*Cos[2*(c + d*x)] + 280*a*b^4*Cos[4*(c + d*x)] - 10
240*a^4*b*Sin[c + d*x] + 8672*a^2*b^3*Sin[c + d*x] + 330*b^5*Sin[c + d*x] + 800*a^2*b^3*Sin[3*(c + d*x)] - 255
*b^5*Sin[3*(c + d*x)] - 105*b^5*Sin[5*(c + d*x)]))/(9240*b^7*d*Sqrt[a + b*Sin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1355\) vs. \(2(498)=996\).

Time = 3.00 (sec) , antiderivative size = 1356, normalized size of antiderivative = 2.91

method	result	size
default	\(\text {Expression too large to display}\)	\(1356\)

[In]

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

[Out]

-2/1155*(5120*((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))*a^6*b-3840*((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^2-6656*((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^3+4392*((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^4+1476*((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^5-552*((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^6+10496*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+si
n(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^2-5868*((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^4+492*((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^6+60*((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^7-5120*((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^7-2368*a^3
*b^4+2560*a^5*b^2+60*a*b^6+140*a*b^6*sin(d*x+c)^6-200*a^2*b^5*sin(d*x+c)^5+320*a^3*b^4*sin(d*x+c)^4-466*a*b^6*
sin(d*x+c)^4-640*a^4*b^3*sin(d*x+c)^3+892*a^2*b^5*sin(d*x+c)^3-2560*a^5*b^2*sin(d*x+c)^2+2048*a^3*b^4*sin(d*x+
c)^2+266*a*b^6*sin(d*x+c)^2+640*a^4*b^3*sin(d*x+c)-692*a^2*b^5*sin(d*x+c)-105*b^7*sin(d*x+c)^7+300*b^7*sin(d*x
+c)^5-255*b^7*sin(d*x+c)^3+60*b^7*sin(d*x+c))/b^8/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 = 0.26 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/3465*(2*(sqrt(2)*(2560*a^6*b - 3648*a^4*b^3 + 984*a^2*b^5 + 45*b^7)*sin(d*x + c) + sqrt(2)*(2560*a^7 - 3648*
a^5*b^2 + 984*a^3*b^4 + 45*a*b^6))*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)*(2560*a^6*b - 3648*a^4*b^3 +
984*a^2*b^5 + 45*b^7)*sin(d*x + c) + sqrt(2)*(2560*a^7 - 3648*a^5*b^2 + 984*a^3*b^4 + 45*a*b^6))*sqrt(-I*b)*w
eierstrassPInverse(-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*s
in(d*x + c) + 2*I*a)/b) - 6*(sqrt(2)*(-1280*I*a^5*b^2 + 1344*I*a^3*b^4 - 123*I*a*b^6)*sin(d*x + c) + sqrt(2)*(
-1280*I*a^6*b + 1344*I*a^4*b^3 - 123*I*a^2*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)) - 6*(sqrt(2)*(1280*I*a^5*b^2 - 1344*I*a^3*b^4 + 123*I*a*b^6)*
sin(d*x + c) + sqrt(2)*(1280*I*a^6*b - 1344*I*a^4*b^3 + 123*I*a^2*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*(140*a*b^6*cos(d*x + c)^5 - 2*(
160*a^3*b^4 - 23*a*b^6)*cos(d*x + c)^3 - 2*(1280*a^5*b^2 - 1344*a^3*b^4 + 123*a*b^6)*cos(d*x + c) - (105*b^7*c
os(d*x + c)^5 - 5*(40*a^2*b^5 + 3*b^7)*cos(d*x + c)^3 + 2*(320*a^4*b^3 - 246*a^2*b^5 - 15*b^7)*cos(d*x + c))*s
in(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^9*d*sin(d*x + c) + a*b^8*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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