3.1168 \(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\)

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

[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]  time = 1.18, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {10 \left (8 a^2-11 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{429 b^3 d}+\frac {4 a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3003 b^4 d}-\frac {8 \left (-683 a^2 b^2+480 a^4+77 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^5 d}+\frac {64 a \left (-118 a^2 b^2+80 a^4+17 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15015 b^6 d}-\frac {8 a \left (-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^7 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (-2048 a^4 b^2+453 a^2 b^4+1280 a^6+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {24 a \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{143 b^2 d}-\frac {2 \sin ^5(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{13 b d} \]

Antiderivative was successfully verified.

[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 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2752

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 2895

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 3023

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 3049

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*} \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\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 ) F\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*}

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Mathematica [A]  time = 5.59, size = 382, normalized size = 0.81 \[ \frac {384 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+3 b \cos (c+d x) \left (81920 a^6+20480 a^5 b \sin (c+d x)-125952 a^4 b^2-28608 a^3 b^3 \sin (c+d x)-1600 a^3 b^3 \sin (3 (c+d x))+23760 a^2 b^4-70 \left (8 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))+\left (5120 a^4 b^2-5792 a^2 b^4-8547 b^6\right ) \cos (2 (c+d x))+2332 a b^5 \sin (c+d x)+1390 a b^5 \sin (3 (c+d x))+210 a b^5 \sin (5 (c+d x))+1155 b^6 \cos (6 (c+d x))+6622 b^6\right )-384 \left (1280 a^7+1280 a^6 b-2048 a^5 b^2-2048 a^4 b^3+453 a^3 b^4+453 a^2 b^5+231 a b^6+231 b^7\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{720720 b^7 d \sqrt {a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[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]])

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fricas [F]  time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right ) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(cos(d*x + c)^6 - cos(d*x + c)^4)*sin(d*x + c)/sqrt(b*sin(d*x + c) + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 1.75, size = 1619, normalized size = 3.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(-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)^4-64
0*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)^2-42*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)-105*
a*b^7*sin(d*x+c)^7+2560*a^6*b^2+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+140*a^2*b^6*sin(d*x
+c)^6+2233*b^8*sin(d*x+c)^4-308*b^8*sin(d*x+c)^2+1155*b^8*sin(d*x+c)^8-3080*b^8*sin(d*x+c)^6-3776*a^4*b^4+544*
a^2*b^6+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)*Ell
ipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^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)*EllipticF(((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)*El
lipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^8-1740*((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*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^4+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)/b^8/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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