\(\int \frac {\cos (c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [339]

Optimal result

Integrand size = 27, antiderivative size = 63 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}} \] Output:

-2*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(1/2)/d+2*cos(d*x+ 
c)/d/(a+a*sin(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.84 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a (1+\sin (c+d x))}} \] Input:

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

Output:

((2*Cos[(c + d*x)/2] - Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[ 
1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Sin[(c + d*x)/2])*(Cos[(c + d 
*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 3353, 3042, 3460, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x) \sqrt {a \sin (c+d x)+a}}dx\)

\(\Big \downarrow \) 3353

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3460

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {2 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{a^2}\)

\(\Big \downarrow \) 3252

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

\(\Big \downarrow \) 219

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

Input:

Int[(Cos[c + d*x]*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

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

Defintions of rubi rules used

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 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 3353

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

rule 3460

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

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40

method	result	size
default	\(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )-\sqrt {a -a \sin \left (d x +c \right )}\right )}{a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(88\)

Input:

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

Output:

-2*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(a^(1/2)*arctanh((a-a*sin(d*x+ 
c))^(1/2)/a^(1/2))-(a-a*sin(d*x+c))^(1/2))/a/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. 236 vs. \(2 (55) = 110\).

Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.75 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{2 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \] Input:

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

Output:

1/2*(sqrt(a)*(cos(d*x + c) + sin(d*x + c) + 1)*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*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a*d*cos(d 
*x + c) + a*d*sin(d*x + c) + a*d)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (55) = 110\).

Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{2 \, d} \] Input:

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

Output:

-1/2*sqrt(2)*sqrt(a)*(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)))/(a*sgn(cos(- 
1/4*pi + 1/2*d*x + 1/2*c))) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)/(a*sgn(cos( 
-1/4*pi + 1/2*d*x + 1/2*c))))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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