\(\int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx\) [384]

Optimal result

Integrand size = 23, antiderivative size = 52 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\cot (c+d x)}{a d} \] Output:

-(a+b)^(1/2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/a^(3/2)/d-cot(d*x+c)/a 
/d
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {-\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )-\sqrt {a} \cot (c+d x)}{a^{3/2} d} \] Input:

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

Output:

(-(Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]) - Sqrt[a]*Cot[c 
 + d*x])/(a^(3/2)*d)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3674, 264, 218}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3674

\(\displaystyle \frac {\int \frac {\cot ^2(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot (c+d x)}{a}}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\cot (c+d x)}{a}}{d}\)

Input:

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

Output:

(-((Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/a^(3/2)) - Cot 
[c + d*x]/a)/d
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 3042

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

rule 3674

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*((d_.)*tan[(e_.) + (f_.) 
*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[f 
f/f   Subst[Int[(d*ff*x)^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(p + 1) 
), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ 
[p]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\frac {\left (-a -b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a \sqrt {a \left (a +b \right )}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\)	\(55\)
default	\(\frac {\frac {\left (-a -b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a \sqrt {a \left (a +b \right )}}-\frac {1}{a \tan \left (d x +c \right )}}{d}\)	\(55\)
risch	\(-\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 a^{2} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 a^{2} d}\)	\(121\)

Input:

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

Output:

1/d*(1/a*(-a-b)/(a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2))-1 
/a/tan(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (44) = 88\).

Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 5.58 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {\sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) - 4 \, \cos \left (d x + c\right )}{4 \, a d \sin \left (d x + c\right )}, \frac {\sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )}{2 \, a d \sin \left (d x + c\right )}\right ] \] Input:

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

Output:

[1/4*(sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^ 
2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + 
 a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^ 
2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin( 
d*x + c) - 4*cos(d*x + c))/(a*d*sin(d*x + c)), 1/2*(sqrt((a + b)/a)*arctan 
(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + 
 c)*sin(d*x + c)))*sin(d*x + c) - 2*cos(d*x + c))/(a*d*sin(d*x + c))]
 

Sympy [F]

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

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

Output:

Integral(cot(c + d*x)**2/(a + b*sin(c + d*x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {{\left (a + b\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a} + \frac {1}{a \tan \left (d x + c\right )}}{d} \] Input:

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

Output:

-((a + b)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/(sqrt((a + b)*a)*a) 
 + 1/(a*tan(d*x + c)))/d
 

Giac [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.63 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a + b\right )}}{\sqrt {a^{2} + a b} a} + \frac {1}{a \tan \left (d x + c\right )}}{d} \] Input:

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

Output:

-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + 
b*tan(d*x + c))/sqrt(a^2 + a*b)))*(a + b)/(sqrt(a^2 + a*b)*a) + 1/(a*tan(d 
*x + c)))/d
 

Mupad [B] (verification not implemented)

Time = 34.64 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\mathrm {cot}\left (c+d\,x\right )}{a\,d}-\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,\sqrt {a+b}}{a^{3/2}\,d} \] Input:

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

Output:

- cot(c + d*x)/(a*d) - (atan((tan(c + d*x)*(a + b)^(1/2))/a^(1/2))*(a + b) 
^(1/2))/(a^(3/2)*d)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 559, normalized size of antiderivative = 10.75 \[ \int \frac {\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {2 \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) \sin \left (d x +c \right )-2 \sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) \sin \left (d x +c \right ) a -2 \sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) \sin \left (d x +c \right ) b -2 \cos \left (d x +c \right ) a^{2}+\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a +\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b -\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a -\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+\sqrt {a}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b}{2 \sin \left (d x +c \right ) a^{3} d} \] Input:

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

Output:

(2*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan( 
(tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin( 
c + d*x) - 2*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + 
d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d*x)*a 
 - 2*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)* 
a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(c + d*x)*b - 2*cos 
(c + d*x)*a**2 + sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - 
a - 2*b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d 
*x)/2))*sin(c + d*x) - sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + 
 b) - a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c 
+ d*x)/2))*sin(c + d*x) + sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*lo 
g( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin 
(c + d*x)*a + sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2* 
sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)*b 
- sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a 
+ b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)*a - sqrt(a)*sqrt( 
2*sqrt(b)*sqrt(a + b) - a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) 
 + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)*b)/(2*sin(c + d*x)*a**3*d)