\(\int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1170]

Optimal result

Integrand size = 29, antiderivative size = 283 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {8 \left (32 a^4-57 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d} \]

[Out]

-2/63*cos(d*x+c)^3*(8*a-7*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^2/d+4/315*cos(d*x+c)*(a*(32*a^2-33*b^2)-3*b*(
8*a^2-7*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^4/d-8/315*(32*a^4-57*a^2*b^2+21*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^5/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+8/315*a*(32*a^4-65*a^2*b^2+33*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)*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*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2944, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d}-\frac {8 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d} \]

[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]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63*b^2*d) + (8*(32*a^4 - 57*a^2*b^2 + 2
1*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d*Sqrt[(a + b*Sin[c + d
*x])/(a + b)]) - (8*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*
Sin[c + d*x])/(a + b)])/(315*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32
*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(315*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 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 ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {4 \int \frac {\cos ^2(c+d x) \left (-\frac {a b}{2}-\frac {1}{2} \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{21 b^2} \\ & = -\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d}+\frac {16 \int \frac {a b \left (2 a^2-3 b^2\right )+\frac {1}{4} \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 b^4} \\ & = -\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d}+\frac {\left (4 \left (32 a^4-57 a^2 b^2+21 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{315 b^5}-\frac {\left (4 a \left (32 a^4-65 a^2 b^2+33 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{315 b^5} \\ & = -\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d}+\frac {\left (4 \left (32 a^4-57 a^2 b^2+21 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}{315 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a \left (32 a^4-65 a^2 b^2+33 b^4\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}{315 b^5 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {8 \left (32 a^4-57 a^2 b^2+21 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)}}{315 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^4-65 a^2 b^2+33 b^4\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}}}{315 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {-32 \left (32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+32 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (-512 a^4+880 a^2 b^2-203 b^4-8 \left (4 a^2 b^2-21 b^4\right ) \cos (2 (c+d x))+35 b^4 \cos (4 (c+d x))-128 a^3 b \sin (c+d x)+202 a b^3 \sin (c+d x)+10 a b^3 \sin (3 (c+d x))\right )}{1260 b^5 d \sqrt {a+b \sin (c+d x)}} \]

[In]

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

[Out]

(-32*(32*a^5 + 32*a^4*b - 57*a^3*b^2 - 57*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/
(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 32*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*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]*(-512*a^4 + 880*a^2*b^2 - 203*b^4 - 8*
(4*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 128*a^3*b*Sin[c + d*x] + 202*a*b^3*Sin[c + d
*x] + 10*a*b^3*Sin[3*(c + d*x)]))/(1260*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. \(1189\) vs. \(2(329)=658\).

Time = 1.22 (sec) , antiderivative size = 1190, normalized size of antiderivative = 4.20

method	result	size
default	\(\text {Expression too large to display}\)	\(1190\)

[In]

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

[Out]

-2/315*(-35*b^6*sin(d*x+c)^6+5*a*b^5*sin(d*x+c)^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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6-356*(
(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^2+312*((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
^4-84*((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^6-128*((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^
5*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)*Elliptic
F(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2+260*((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^3-180*((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))*a^2*b^4-132*((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^5+84*((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^6-8*a^2*b^4*sin(d*x+c)^4+112*b^6*sin(d*x+
c)^4+16*a^3*b^3*sin(d*x+c)^3-34*a*b^5*sin(d*x+c)^3+64*a^4*b^2*sin(d*x+c)^2-98*a^2*b^4*sin(d*x+c)^2-77*b^6*sin(
d*x+c)^2-16*a^3*b^3*sin(d*x+c)+29*a*b^5*sin(d*x+c)-64*a^4*b^2+106*a^2*b^4)/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 = 0.15 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {2} {\left (32 \, a^{5} - 69 \, a^{3} b^{2} + 39 \, a b^{4}\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 ) + 4 \, \sqrt {2} {\left (32 \, a^{5} - 69 \, a^{3} b^{2} + 39 \, a b^{4}\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^{4} b - 57 i \, a^{2} b^{3} + 21 i \, 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^{4} b + 57 i \, a^{2} b^{3} - 21 i \, 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 (40 \, a b^{4} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{3} b^{2} - 33 \, a b^{4}\right )} \cos \left (d x + c\right ) - {\left (35 \, b^{5} \cos \left (d x + c\right )^{3} - 6 \, {\left (8 \, a^{2} b^{3} - 7 \, 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 )}}{945 \, 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]

-2/945*(4*sqrt(2)*(32*a^5 - 69*a^3*b^2 + 39*a*b^4)*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) + 4*sqrt(2)*(32*a^5 - 69*
a^3*b^2 + 39*a*b^4)*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^4*b - 57*I*a^2*b^3 + 21*I*b^5)*sqr
t(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^4*b + 57*I*a^2*b^3 - 21*I*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*(40*a*b^4*cos(d*x + c)^3 - 2*(32*a^3*b^2 - 33*a*b^4)*c
os(d*x + c) - (35*b^5*cos(d*x + c)^3 - 6*(8*a^2*b^3 - 7*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. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]

[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]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[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)