\(\int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [326]

Optimal result

Integrand size = 23, antiderivative size = 89 \[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {3 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \sqrt {a+a \sin (c+d x)}}{d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(206\) vs. \(2(89)=178\).

Time = 0.98 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.31 \[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-4 \cos \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {3}{2} (c+d x)\right )+4 \sin \left (\frac {1}{2} (c+d x)\right )-\log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+\log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+2 \sin \left (\frac {3}{2} (c+d x)\right )\right )}{d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )-\sec \left (\frac {1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )+\sec \left (\frac {1}{4} (c+d x)\right )\right )} \] Input:

Integrate[Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-4*Cos[(c + d*x)/2] + 2*Co 
s[(3*(c + d*x))/2] + 4*Sin[(c + d*x)/2] - Log[1 + Cos[(c + d*x)/2] - Sin[( 
c + d*x)/2]]*Sin[c + d*x] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*S 
in[c + d*x] + 2*Sin[(3*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + 
 d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4]))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3195, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3195

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3460

\(\displaystyle \frac {a \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx+\frac {6 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{2 a}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {6 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{2 a}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {6 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^2 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {6 a^2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}-\frac {\cot (c+d x) \sqrt {a \sin (c+d x)+a}}{d}\)

Input:

Int[Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

-((Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d) + ((-2*a^(3/2)*ArcTanh[(Sqrt[ 
a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (6*a^2*Cos[c + d*x])/(d*Sq 
rt[a + a*Sin[c + d*x]]))/(2*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 3195

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)/tan[(e_.) + (f_.)*(x_)]^2, 
 x_Symbol] :> Simp[-(a + b*Sin[e + f*x])^m/(f*Tan[e + f*x]), x] + Simp[1/a 
  Int[(a + b*Sin[e + f*x])^m*((b*m - a*(m + 1)*Sin[e + f*x])/Sin[e + f*x]), 
 x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1 
/2] &&  !LtQ[m, -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 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 = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.45

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (2 \sqrt {a -a \sin \left (d x +c \right )}\, \sin \left (d x +c \right ) a^{\frac {3}{2}}-\sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )\right )}{\sin \left (d x +c \right ) a^{\frac {3}{2}} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(129\)

Input:

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

Output:

(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(2*(a-a*sin(d*x+c))^(1/2)*sin(d*x 
+c)*a^(3/2)-(a-a*sin(d*x+c))^(1/2)*a^(3/2)-arctanh((a-a*sin(d*x+c))^(1/2)/ 
a^(1/2))*a^2*sin(d*x+c))/sin(d*x+c)/a^(3/2)/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. 279 vs. \(2 (79) = 158\).

Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.13 \[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {{\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{4 \, {\left (d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \] Input:

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

Output:

1/4*((cos(d*x + c)^2 - (cos(d*x + c) + 1)*sin(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*(2*cos(d*x + c)^2 + (2*cos(d*x + c) + 3)*sin(d*x 
+ c) - cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^2 - (d* 
cos(d*x + c) + d)*sin(d*x + c) - d)
 

Sympy [F]

\[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cot ^{2}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a*(sin(c + d*x) + 1))*cot(c + d*x)**2, x)
 

Maxima [F]

\[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(a*sin(d*x + c) + a)*cot(d*x + c)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.67 \[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}\right )} \sqrt {a}}{4 \, d} \] Input:

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

Output:

-1/4*sqrt(2)*(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)) + 8*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2* 
d*x + 1/2*c) + 4*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))*sqrt(a)/d
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \cot ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{2}d x \right ) \] Input:

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

Output:

sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**2,x)