\(\int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) [1144]

Optimal result

Integrand size = 29, antiderivative size = 332 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {8 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d} \]

[Out]

-2/11*cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d-2/693*cos(d*x+c)^3*(8*a^2-9*b^2-7*a*b*sin(d*x+c))*(a+b*sin(d*x+c))
^(1/2)/b^2/d+4/3465*cos(d*x+c)*(32*a^4-69*a^2*b^2+45*b^4-24*a*b*(a^2-2*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)
/b^4/d-8/3465*a*(32*a^4-93*a^2*b^2+93*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)*Ellip
ticE(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^5/d/((a+b*sin(d*x+c))/(a+b))^
(1/2)+8/3465*(32*a^6-101*a^4*b^2+114*a^2*b^4-45*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^5/d/(a+b*si
n(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2941, 2944, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{3465 b^4 d}+\frac {8 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d} \]

[In]

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

[Out]

(-2*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + (8*a*(32*a^4 - 93*a^2*b^2 + 93*b^4)*EllipticE[(c - Pi/2
+ d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*(32*a^
6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])
/(a + b)])/(3465*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^2 - 9*b^2 -
7*a*b*Sin[c + d*x]))/(693*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 - 69*a^2*b^2 + 45*b^4 - 2
4*a*b*(a^2 - 2*b^2)*Sin[c + d*x]))/(3465*b^4*d)

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 2941

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

Rule 2944

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {2}{11} \int \frac {\cos ^4(c+d x) \left (\frac {b}{2}+\frac {1}{2} a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {8 \int \frac {\cos ^2(c+d x) \left (-\frac {1}{4} b \left (a^2-9 b^2\right )-2 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{231 b^2} \\ & = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac {32 \int \frac {\frac {1}{8} b \left (8 a^4-21 a^2 b^2+45 b^4\right )+\frac {1}{8} a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^4} \\ & = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac {\left (4 a \left (32 a^4-93 a^2 b^2+93 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3465 b^5}-\frac {\left (4 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^5} \\ & = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac {\left (4 a \left (32 a^4-93 a^2 b^2+93 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}{3465 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 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}{3465 b^5 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {8 a \left (32 a^4-93 a^2 b^2+93 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)}}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 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}}}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.98 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {-64 a \left (32 a^5+32 a^4 b-93 a^3 b^2-93 a^2 b^3+93 a b^4+93 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+64 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (1024 a^5-2912 a^3 b^2+748 a b^4+16 \left (4 a^3 b^2-183 a b^4\right ) \cos (2 (c+d x))-700 a b^4 \cos (4 (c+d x))+256 a^4 b \sin (c+d x)-692 a^2 b^3 \sin (c+d x)+990 b^5 \sin (c+d x)-20 a^2 b^3 \sin (3 (c+d x))-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )}{27720 b^5 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(-64*a*(32*a^5 + 32*a^4*b - 93*a^3*b^2 - 93*a^2*b^3 + 93*a*b^4 + 93*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b
)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 64*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*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]*(1024*a^5 - 2912*a^3*b^
2 + 748*a*b^4 + 16*(4*a^3*b^2 - 183*a*b^4)*Cos[2*(c + d*x)] - 700*a*b^4*Cos[4*(c + d*x)] + 256*a^4*b*Sin[c + d
*x] - 692*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] - 20*a^2*b^3*Sin[3*(c + d*x)] - 765*b^5*Sin[3*(c + d*x)]
- 315*b^5*Sin[5*(c + d*x)]))/(27720*b^5*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. \(2(374)=748\).

Time = 2.10 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.08

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

[In]

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

[Out]

2/3465*(128*((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
ipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b-96*((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^2-404*((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+288*((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+456*((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^5-192*((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*b^6+500*((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^2-744*((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+372*((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^6-180*((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^7-128*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s
in(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^7-178*a^3*b^4+64*a^5
*b^2-180*a*b^6+350*a*b^6*sin(d*x+c)^6-5*a^2*b^5*sin(d*x+c)^5+8*a^3*b^4*sin(d*x+c)^4-1066*a*b^6*sin(d*x+c)^4-16
*a^4*b^3*sin(d*x+c)^3+52*a^2*b^5*sin(d*x+c)^3-64*a^5*b^2*sin(d*x+c)^2+170*a^3*b^4*sin(d*x+c)^2+896*a*b^6*sin(d
*x+c)^2+16*a^4*b^3*sin(d*x+c)-47*a^2*b^5*sin(d*x+c)+315*b^7*sin(d*x+c)^7-900*b^7*sin(d*x+c)^5+765*b^7*sin(d*x+
c)^3-180*b^7*sin(d*x+c))/b^6/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 = 0.14 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.76 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (64 \, a^{6} - 210 \, a^{4} b^{2} + 249 \, a^{2} b^{4} - 135 \, 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 ) + 2 \, \sqrt {2} {\left (64 \, a^{6} - 210 \, a^{4} b^{2} + 249 \, a^{2} b^{4} - 135 \, 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 (32 i \, a^{5} b - 93 i \, a^{3} b^{3} + 93 i \, a b^{5}\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 (-32 i \, a^{5} b + 93 i \, a^{3} b^{3} - 93 i \, a b^{5}\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 (315 \, b^{6} \cos \left (d x + c\right )^{5} + 5 \, {\left (8 \, a^{2} b^{4} - 9 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{4} b^{2} - 69 \, a^{2} b^{4} + 45 \, b^{6}\right )} \cos \left (d x + c\right ) - {\left (35 \, a b^{5} \cos \left (d x + c\right )^{3} - 48 \, {\left (a^{3} b^{3} - 2 \, 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 )}}{10395 \, b^{6} d} \]

[In]

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

[Out]

-2/10395*(2*sqrt(2)*(64*a^6 - 210*a^4*b^2 + 249*a^2*b^4 - 135*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)*(64*a^6 - 210*a^4*b^2 + 249*a^2*b^4 - 135*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)*(32*I*a^5*b
- 93*I*a^3*b^3 + 93*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)) + 6*sqrt(2)*(-32*I*a^5*b + 93*I*a^3*b^3 - 93*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, -8/2
7*(-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*(315*b^6*cos(d*x + c
)^5 + 5*(8*a^2*b^4 - 9*b^6)*cos(d*x + c)^3 - 2*(32*a^4*b^2 - 69*a^2*b^4 + 45*b^6)*cos(d*x + c) - (35*a*b^5*cos
(d*x + c)^3 - 48*(a^3*b^3 - 2*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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