\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1168]

Optimal result

Integrand size = 31, antiderivative size = 471 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}+\frac {8 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15015 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15015 b^7 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

64/15015*a*(80*a^4-118*a^2*b^2+17*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^6/d-8/15015*(480*a^4-683*a^2*b^2+77
*b^4)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+4/3003*a*(160*a^2-223*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a
+b*sin(d*x+c))^(1/2)/b^4/d-10/429*(8*a^2-11*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/b^3/d+24/143*a
*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(1/2)/b^2/d-2/13*cos(d*x+c)*sin(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/b/d-
8/15015*(1280*a^6-2048*a^4*b^2+453*a^2*b^4+231*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^7/d/((a+b*sin(d*x+c)
)/(a+b))^(1/2)+8/15015*a*(1280*a^6-2368*a^4*b^2+875*a^2*b^4+213*b^6)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1
/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/
2)/b^7/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 471, 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 = {2974, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {4 a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}-\frac {8 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{15015 b^7 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {24 a \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \sin ^5(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d} \]

[In]

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

[Out]

(64*a*(80*a^4 - 118*a^2*b^2 + 17*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^6*d) - (8*(480*a^4 - 683
*a^2*b^2 + 77*b^4)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^5*d) + (4*a*(160*a^2 - 223*b^2
)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3003*b^4*d) - (10*(8*a^2 - 11*b^2)*Cos[c + d*x]*Sin[c
+ d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(429*b^3*d) + (24*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/
(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(13*b*d) + (8*(1280*a^6 - 2048*a^4*b^2
+ 453*a^2*b^4 + 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^7*d*S
qrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(1280*a^6 - 2368*a^4*b^2 + 875*a^2*b^4 + 213*b^6)*EllipticF[(c - Pi/
2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*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 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 {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {4 \int \frac {\sin ^3(c+d x) \left (\frac {1}{4} \left (96 a^2-143 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {15}{4} \left (8 a^2-11 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{143 b^2} \\ & = -\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {8 \int \frac {\sin ^2(c+d x) \left (-\frac {45}{4} a \left (8 a^2-11 b^2\right )+\frac {3}{2} b \left (2 a^2-11 b^2\right ) \sin (c+d x)+\frac {3}{4} a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{1287 b^3} \\ & = \frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {16 \int \frac {\sin (c+d x) \left (\frac {3}{2} a^2 \left (160 a^2-223 b^2\right )-15 a b \left (a^2-b^2\right ) \sin (c+d x)-\frac {3}{4} \left (480 a^4-683 a^2 b^2+77 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9009 b^4} \\ & = -\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {32 \int \frac {-\frac {3}{4} a \left (480 a^4-683 a^2 b^2+77 b^4\right )+\frac {3}{8} b \left (160 a^4-181 a^2 b^2-231 b^4\right ) \sin (c+d x)+9 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^5} \\ & = \frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {64 \int \frac {-\frac {9}{8} a b \left (160 a^4-211 a^2 b^2+9 b^4\right )-\frac {9}{16} \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{135135 b^6} \\ & = \frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}-\frac {\left (4 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15015 b^7}+\frac {\left (4 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15015 b^7} \\ & = \frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}+\frac {\left (4 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 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}{15015 b^7 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 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}{15015 b^7 \sqrt {a+b \sin (c+d x)}} \\ & = \frac {64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {24 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \cos (c+d x) \sin ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d}+\frac {8 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 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)}}{15015 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 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}}}{15015 b^7 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.24 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {-384 \left (1280 a^7+1280 a^6 b-2048 a^5 b^2-2048 a^4 b^3+453 a^3 b^4+453 a^2 b^5+231 a b^6+231 b^7\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+384 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+3 b \cos (c+d x) \left (81920 a^6-125952 a^4 b^2+23760 a^2 b^4+6622 b^6+\left (5120 a^4 b^2-5792 a^2 b^4-8547 b^6\right ) \cos (2 (c+d x))-70 \left (8 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))+1155 b^6 \cos (6 (c+d x))+20480 a^5 b \sin (c+d x)-28608 a^3 b^3 \sin (c+d x)+2332 a b^5 \sin (c+d x)-1600 a^3 b^3 \sin (3 (c+d x))+1390 a b^5 \sin (3 (c+d x))+210 a b^5 \sin (5 (c+d x))\right )}{720720 b^7 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(-384*(1280*a^7 + 1280*a^6*b - 2048*a^5*b^2 - 2048*a^4*b^3 + 453*a^3*b^4 + 453*a^2*b^5 + 231*a*b^6 + 231*b^7)*
EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 384*a*(1280*a^6 - 2368*a^
4*b^2 + 875*a^2*b^4 + 213*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a +
b)] + 3*b*Cos[c + d*x]*(81920*a^6 - 125952*a^4*b^2 + 23760*a^2*b^4 + 6622*b^6 + (5120*a^4*b^2 - 5792*a^2*b^4 -
8547*b^6)*Cos[2*(c + d*x)] - 70*(8*a^2*b^4 - 11*b^6)*Cos[4*(c + d*x)] + 1155*b^6*Cos[6*(c + d*x)] + 20480*a^5
*b*Sin[c + d*x] - 28608*a^3*b^3*Sin[c + d*x] + 2332*a*b^5*Sin[c + d*x] - 1600*a^3*b^3*Sin[3*(c + d*x)] + 1390*
a*b^5*Sin[3*(c + d*x)] + 210*a*b^5*Sin[5*(c + d*x)]))/(720720*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. \(1618\) vs. \(2(501)=1002\).

Time = 1.82 (sec) , antiderivative size = 1619, normalized size of antiderivative = 3.44

method	result	size
default	\(\text {Expression too large to display}\)	\(1619\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15015*(-924*((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^8-5120*((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^8+924*((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^8+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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))
^(1/2))*a^7*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^6*b^2-9472*((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^3+6504*((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^4+3500*((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^5-1740*((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^6+852*((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^7+13312*((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*b^2-10004*((
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*s
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4+888*((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^
6-105*a*b^7*sin(d*x+c)^7+140*a^2*b^6*sin(d*x+c)^6-200*a^3*b^5*sin(d*x+c)^5+410*a*b^7*sin(d*x+c)^5+320*a^4*b^4*
sin(d*x+c)^4-642*a^2*b^6*sin(d*x+c)^4-640*a^5*b^3*sin(d*x+c)^3+1244*a^3*b^5*sin(d*x+c)^3-541*a*b^7*sin(d*x+c)^
3-2560*a^6*b^2*sin(d*x+c)^2+3456*a^4*b^4*sin(d*x+c)^2+2560*a^6*b^2-3776*a^4*b^4+544*a^2*b^6-42*a^2*b^6*sin(d*x
+c)^2+640*a^5*b^3*sin(d*x+c)-1044*a^3*b^5*sin(d*x+c)+236*a*b^7*sin(d*x+c)+1155*b^8*sin(d*x+c)^8-3080*b^8*sin(d
*x+c)^6+2233*b^8*sin(d*x+c)^4-308*b^8*sin(d*x+c)^2)/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 = 0.19 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \, {\left (8 \, \sqrt {2} {\left (640 \, a^{7} - 1264 \, a^{5} b^{2} + 543 \, a^{3} b^{4} + 102 \, a b^{6}\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 ) + 8 \, \sqrt {2} {\left (640 \, a^{7} - 1264 \, a^{5} b^{2} + 543 \, a^{3} b^{4} + 102 \, a b^{6}\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 \, \sqrt {2} {\left (1280 i \, a^{6} b - 2048 i \, a^{4} b^{3} + 453 i \, a^{2} b^{5} + 231 i \, 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 ) + 6 \, \sqrt {2} {\left (-1280 i \, a^{6} b + 2048 i \, a^{4} b^{3} - 453 i \, a^{2} b^{5} - 231 i \, 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 (1260 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (160 \, a^{3} b^{4} + 29 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (1280 \, a^{5} b^{2} - 1088 \, a^{3} b^{4} - 213 \, a b^{6}\right )} \cos \left (d x + c\right ) - {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (40 \, a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (320 \, a^{4} b^{3} - 222 \, a^{2} b^{5} - 77 \, 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 )}}{45045 \, b^{8} d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-2/45045*(8*sqrt(2)*(640*a^7 - 1264*a^5*b^2 + 543*a^3*b^4 + 102*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) + 8*s
qrt(2)*(640*a^7 - 1264*a^5*b^2 + 543*a^3*b^4 + 102*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) + 6*sqrt(2)*(128
0*I*a^6*b - 2048*I*a^4*b^3 + 453*I*a^2*b^5 + 231*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)) + 6*sqrt(2)*(-1280*I*a^6*b + 2048*I*a^4*b^3 - 453*I*a^
2*b^5 - 231*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, weie
rstrassPInverse(-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*(1260*a*b^6*cos(d*x + c)^5 - 10*(160*a^3*b^4 + 29*a*b^6)*cos(d*x + c)^3 + 2*(1280*a^
5*b^2 - 1088*a^3*b^4 - 213*a*b^6)*cos(d*x + c) - (1155*b^7*cos(d*x + c)^5 - 35*(40*a^2*b^5 + 11*b^7)*cos(d*x +
c)^3 + 6*(320*a^4*b^3 - 222*a^2*b^5 - 77*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^8*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

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

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(1/2),x)

[Out]

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(1/2), x)