\(\int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1174]

Optimal result

Integrand size = 29, antiderivative size = 261 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {8 a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d} \] Output:

8/35*a*(32*a^2-29*b^2)*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)+8/35 
*(32*a^4-37*a^2*b^2+5*b^4)*InverseJacobiAM(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) 
+2/7*cos(d*x+c)^3*(8*a+b*sin(d*x+c))/b^2/d/(a+b*sin(d*x+c))^(1/2)-4/35*cos 
(d*x+c)*(a+b*sin(d*x+c))^(1/2)*(32*a^2-5*b^2-24*a*b*sin(d*x+c))/b^4/d
 

Mathematica [A] (verified)

Time = 4.32 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {16 a \left (32 a^3+32 a^2 b-29 a b^2-29 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-16 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-256 a^3+216 a b^2-16 a b^2 \cos (2 (c+d x))+\left (-64 a^2 b+45 b^3\right ) \sin (c+d x)+5 b^3 \sin (3 (c+d x))\right )}{70 b^5 d \sqrt {a+b \sin (c+d x)}} \] Input:

Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

(16*a*(32*a^3 + 32*a^2*b - 29*a*b^2 - 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x 
)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 16*(32*a^4 - 37*a 
^2*b^2 + 5*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]*(-256*a^3 + 216*a*b^2 - 16*a*b^2 
*Cos[2*(c + d*x)] + (-64*a^2*b + 45*b^3)*Sin[c + d*x] + 5*b^3*Sin[3*(c + d 
*x)]))/(70*b^5*d*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3342, 27, 3042, 3344, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sin (c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^4}{(a+b \sin (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3342

\(\displaystyle \frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {12 \int -\frac {\cos ^2(c+d x) (b+8 a \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx}{7 b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \int \frac {\cos ^2(c+d x) (b+8 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \int \frac {\cos (c+d x)^2 (b+8 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3344

\(\displaystyle \frac {6 \left (\frac {4 \int -\frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \left (-\frac {2 \int \frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (-\frac {2 \int \frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {6 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\)

Input:

Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

(2*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(7*b^2*d*Sqrt[a + b*Sin[c + d*x] 
]) + (6*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a* 
b*Sin[c + d*x]))/(15*b^2*d) - (2*((2*a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi 
/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin 
[c + d*x])/(a + b)]) - (2*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/ 
2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*d*Sqrt[a 
 + b*Sin[c + d*x]])))/(15*b^2)))/(7*b^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[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 3342

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*C 
os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p 
 + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( 
p - 1)/(b^2*(m + 1)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin 
[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x 
], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
 

rule 3344

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] + Simp[g^2*( 
(p - 1)/(b^2*(m + p)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si 
n[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs. \(2(246)=492\).

Time = 2.52 (sec) , antiderivative size = 943, normalized size of antiderivative = 3.61

method	result	size
default	\(-\frac {2 \left (-5 b^{5} \sin \left (d x +c \right )^{5}+128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b -96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-148 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}+96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+20 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}+244 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-116 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+8 a \,b^{4} \sin \left (d x +c \right )^{4}-16 a^{2} b^{3} \sin \left (d x +c \right )^{3}+20 b^{5} \sin \left (d x +c \right )^{3}-64 a^{3} b^{2} \sin \left (d x +c \right )^{2}+42 a \,b^{4} \sin \left (d x +c \right )^{2}+16 a^{2} b^{3} \sin \left (d x +c \right )-15 b^{5} \sin \left (d x +c \right )+64 a^{3} b^{2}-50 a \,b^{4}\right )}{35 b^{6} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\)	\(943\)

Input:

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

Output:

-2/35*(-5*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)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d* 
x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b-96*((a+b*sin(d*x+c))/(a-b))^ 
(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Elli 
pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2-148*((a+ 
b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+ 
c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1 
/2))*a^2*b^3+96*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^( 
1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1 
/2),((a-b)/(a+b))^(1/2))*a*b^4+20*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d* 
x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin 
(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^5-128*((a+b*sin(d*x+c))/(a-b) 
)^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*El 
lipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5+244*((a+b* 
sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c) 
)/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2 
))*a^3*b^2-116*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1 
/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/ 
2),((a-b)/(a+b))^(1/2))*a*b^4+8*a*b^4*sin(d*x+c)^4-16*a^2*b^3*sin(d*x+c)^3 
+20*b^5*sin(d*x+c)^3-64*a^3*b^2*sin(d*x+c)^2+42*a*b^4*sin(d*x+c)^2+16*a...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.40 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/105*(4*(64*a^5 - 82*a^3*b^2 + 15*a*b^4 + (64*a^4*b - 82*a^2*b^3 + 15*b^5 
)*sin(d*x + c))*sqrt(1/2*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*(64*a^5 - 82*a^3*b^2 + 15*a*b^4 + (64*a^4*b - 82*a^2*b 
^3 + 15*b^5)*sin(d*x + c))*sqrt(-1/2*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) - 12*(-32*I*a^4*b + 29*I*a^2*b^3 + (-32*I*a^3 
*b^2 + 29*I*a*b^4)*sin(d*x + c))*sqrt(1/2*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)) - 12*(32*I*a^4*b - 29*I*a^2*b^3 + (32*I 
*a^3*b^2 - 29*I*a*b^4)*sin(d*x + c))*sqrt(-1/2*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*(8*a*b^4*cos(d*x + c)^3 + 2* 
(32*a^3*b^2 - 29*a*b^4)*cos(d*x + c) - (5*b^5*cos(d*x + c)^3 - 2*(8*a^2*b^ 
3 - 5*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^7*d*si 
n(d*x + c) + a*b^6*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}{\sin \left (d x +c \right )^{2} b^{2}+2 \sin \left (d x +c \right ) a b +a^{2}}d x \] Input:

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

Output:

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