\(\int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [447]

Optimal result

Integrand size = 29, antiderivative size = 156 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {13 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d} \] Output:

13/4*a^(1/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-2/3*a*co 
s(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-1/4*a*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2 
)-2/3*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d-1/2*cot(d*x+c)*csc(d*x+c)*(a+a*s 
in(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.90 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-26 \cos \left (\frac {1}{2} (c+d x)\right )-14 \cos \left (\frac {3}{2} (c+d x)\right )+12 \cos \left (\frac {5}{2} (c+d x)\right )+4 \cos \left (\frac {7}{2} (c+d x)\right )+39 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-39 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-39 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+39 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+26 \sin \left (\frac {1}{2} (c+d x)\right )-14 \sin \left (\frac {3}{2} (c+d x)\right )-12 \sin \left (\frac {5}{2} (c+d x)\right )+4 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{12 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \] Input:

Integrate[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-26*Cos[(c + d*x)/2] - 14* 
Cos[(3*(c + d*x))/2] + 12*Cos[(5*(c + d*x))/2] + 4*Cos[(7*(c + d*x))/2] + 
39*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39*Cos[2*(c + d*x)]*Log[ 
1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39*Log[1 - Cos[(c + d*x)/2] + S 
in[(c + d*x)/2]] + 39*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + 
 d*x)/2]] + 26*Sin[(c + d*x)/2] - 14*Sin[(3*(c + d*x))/2] - 12*Sin[(5*(c + 
 d*x))/2] + 4*Sin[(7*(c + d*x))/2]))/(12*d*(1 + Cot[(c + d*x)/2])*(Csc[(c 
+ d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 3360, 3042, 3230, 3042, 3125, 3523, 27, 3042, 3459, 3042, 3252, 219}

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.

\(\displaystyle \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3360

\(\displaystyle \int \sin (c+d x) \sqrt {\sin (c+d x) a+a}dx+\int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} \left (1-2 \sin ^2(c+d x)\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x) \sqrt {\sin (c+d x) a+a}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx\)

\(\Big \downarrow \) 3230

\(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx+\frac {1}{3} \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \sqrt {\sin (c+d x) a+a}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3125

\(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\)

\(\Big \downarrow \) 3523

\(\displaystyle \frac {\int \frac {1}{2} \csc ^2(c+d x) (a-7 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{2 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \csc ^2(c+d x) (a-7 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(a-7 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {-\frac {13}{2} a \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {13}{2} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a^2 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {13 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a^2 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {13 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a^2 \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{4 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d}\)

Input:

Int[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(-2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[ 
a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + 
 d*x]])/(2*d) + ((13*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin 
[c + d*x]]])/d - (a^2*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/(4*a)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

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

rule 3125

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos 
[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[a^2 - b^2, 0]
 

rule 3230

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

rule 3252

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f)   Subst[Int[1/(b*c + a*d - d*x^2), 
x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3360

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4   Int[(d*Sin[e 
 + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a 
 + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, 
 n}, x] && EqQ[a^2 - b^2, 0] &&  !IGtQ[m, 0]
 

rule 3459

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) 
*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* 
c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d))   Int[Sqrt[a + b*Sin[e + f*x] 
]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x 
] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 
-1]
 

rule 3523

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + 
 f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - 
d^2))   Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a 
*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* 
(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, 
 e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - 
d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14

method	result	size
default	\(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (24 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}} \sin \left (d x +c \right )^{2}-8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \sqrt {a}-39 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{2} a^{2}+15 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}}-9 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\right )}{12 a^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(178\)

Input:

int(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

-1/12*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(24*(-a*(sin(d*x+c)-1))^(1/ 
2)*a^(3/2)*sin(d*x+c)^2-8*(-a*(sin(d*x+c)-1))^(3/2)*sin(d*x+c)^2*a^(1/2)-3 
9*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+c)^2*a^2+15*(-a*(sin( 
d*x+c)-1))^(1/2)*a^(3/2)-9*(-a*(sin(d*x+c)-1))^(3/2)*a^(1/2))/a^(3/2)/sin( 
d*x+c)^2/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
                                                                                    
                                                                                    
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (132) = 264\).

Time = 0.09 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.30 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {39 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (8 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )^{2} + {\left (8 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 17 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right ) - 22 \, \cos \left (d x + c\right ) - 5\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{48 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \] Input:

integrate(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="fri 
cas")
 

Output:

1/48*(39*(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + 
 c) - cos(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 
 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 
3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 
 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^ 
2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(8*cos(d*x 
+ c)^4 + 16*cos(d*x + c)^3 - 9*cos(d*x + c)^2 + (8*cos(d*x + c)^3 - 8*cos( 
d*x + c)^2 - 17*cos(d*x + c) + 5)*sin(d*x + c) - 22*cos(d*x + c) - 5)*sqrt 
(a*sin(d*x + c) + a))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c 
) + (d*cos(d*x + c)^2 - d)*sin(d*x + c) - d)
 

Sympy [F]

\[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a*(sin(c + d*x) + 1))*cos(c + d*x)*cot(c + d*x)**3, x)
 

Maxima [F]

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

integrate(cos(d*x+c)*cot(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="max 
ima")
 

Output:

integrate(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)*cot(d*x + c)^3, x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.35 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (64 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 39 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 96 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, {\left (6 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}\right )} \sqrt {a}}{48 \, d} \] Input:

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

Output:

-1/48*sqrt(2)*(64*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d* 
x + 1/2*c)^3 - 39*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1 
/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1/4*pi 
+ 1/2*d*x + 1/2*c)) - 96*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 
 1/2*d*x + 1/2*c) + 12*(6*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi 
+ 1/2*d*x + 1/2*c)^3 - 5*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 
 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^2)*sqrt(a)/d
 

Mupad [F(-1)]

Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}d x \right ) \] Input:

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

Output:

sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cos(c + d*x)*cot(c + d*x)**3,x)