3.1144 \(\int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=332 \[ -\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{3465 b^4 d}+\frac {8 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d} \]

[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)

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Rubi [A]  time = 0.63, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+32 a^4+45 b^4\right )}{3465 b^4 d}-\frac {8 \left (-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \left (-93 a^2 b^2+32 a^4+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d} \]

Antiderivative was successfully verified.

[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 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 2862

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] && Gt
Q[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 2865

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*} \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 {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 ) 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}}}{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*}

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Mathematica [A]  time = 4.12, size = 326, normalized size = 0.98 \[ \frac {64 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+b \cos (c+d x) \left (1024 a^5+256 a^4 b \sin (c+d x)-2912 a^3 b^2+16 \left (4 a^3 b^2-183 a b^4\right ) \cos (2 (c+d x))-692 a^2 b^3 \sin (c+d x)-20 a^2 b^3 \sin (3 (c+d x))-700 a b^4 \cos (4 (c+d x))+748 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )-64 a \left (32 a^5+32 a^4 b-93 a^3 b^2-93 a^2 b^3+93 a b^4+93 b^5\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{27720 b^5 d \sqrt {a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[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]])

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 1.94, size = 1356, normalized size = 4.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(-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)*Ell
ipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2+350*a*b^6*sin(d*x+c)^6-180*a*b^6+64*a^5*b^2
-178*a^3*b^4-5*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)-1
80*((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+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^7-4
04*((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+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+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)*Ellipt
icF(((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+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
)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b-192*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si
n(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)/b^6/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 \[ \int \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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