\(\int (a+b \cot ^2(c+d x))^{3/2} \, dx\) [32]

Optimal result

Integrand size = 16, antiderivative size = 126 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=-\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {(3 a-2 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\frac {2 (a-b)^{3/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \cot ^2(c+d x)-\cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{\sqrt {a-b}}\right )-b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}+(3 a-2 b) \sqrt {b} \log \left (-\sqrt {b} \cot (c+d x)+\sqrt {a+b \cot ^2(c+d x)}\right )}{2 d} \] Input:

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

Output:

(2*(a - b)^(3/2)*ArcTan[(Sqrt[b] + Sqrt[b]*Cot[c + d*x]^2 - Cot[c + d*x]*S 
qrt[a + b*Cot[c + d*x]^2])/Sqrt[a - b]] - b*Cot[c + d*x]*Sqrt[a + b*Cot[c 
+ d*x]^2] + (3*a - 2*b)*Sqrt[b]*Log[-(Sqrt[b]*Cot[c + d*x]) + Sqrt[a + b*C 
ot[c + d*x]^2]])/(2*d)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4144, 318, 398, 224, 219, 291, 216}

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.

\(\displaystyle \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\)

\(\Big \downarrow \) 4144

\(\displaystyle -\frac {\int \frac {\left (b \cot ^2(c+d x)+a\right )^{3/2}}{\cot ^2(c+d x)+1}d\cot (c+d x)}{d}\)

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {1}{2} \left (2 (a-b)^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)+\sqrt {b} (3 a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{d}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {\frac {1}{2} \left (2 (a-b)^2 \int \frac {1}{1-\frac {(b-a) \cot ^2(c+d x)}{b \cot ^2(c+d x)+a}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a}}+\sqrt {b} (3 a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\frac {1}{2} \left (2 (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )+\sqrt {b} (3 a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

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

rule 3042

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

rule 4144

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> 
With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f)   Subst[Int[(a + b* 
(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, 
 b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || 
EqQ[n^2, 16])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(108)=216\).

Time = 0.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.37

method	result	size
derivativedivides	\(-\frac {b \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}-\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}+\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(298\)
default	\(-\frac {b \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}-\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}+\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(298\)

Input:

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

Output:

-1/2*b*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)/d-3/2/d*b^(1/2)*a*ln(b^(1/2)*co 
t(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2))-1/d*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2* 
(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+1/d*b^(3/2)*l 
n(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2))+2/d*a/b*(b^4*(a-b))^(1/2)/( 
a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c 
))-1/d*a^2*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/ 
(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (108) = 216\).

Time = 0.13 (sec) , antiderivative size = 1095, normalized size of antiderivative = 8.69 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm="fricas")
 

Output:

[-1/4*(2*(a - b)*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) - sqrt(-a + b) 
*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x 
 + 2*c) + b)*sin(2*d*x + 2*c) + (3*a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*d 
*x + 2*c) - 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 
 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x 
+ 2*c) + 2*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b 
)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), 1/2*((3*a - 2*b)*sqrt(-b) 
*arctan(-sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c 
) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - a - b))*sin(2*d*x + 2 
*c) - (a - b)*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) - sqrt(-a + b)*sq 
rt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 
2*c) + b)*sin(2*d*x + 2*c) - (b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2* 
d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), -1/4*(4 
*(a - b)^(3/2)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b) 
/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - a - 
b))*sin(2*d*x + 2*c) + (3*a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) 
 - 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1 
))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + 
2*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2* 
d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), -1/2*(2*(a - b)^(3/2)*arctan(-s...
 

Sympy [F]

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

integrate((a+b*cot(d*x+c)**2)**(3/2),x)
 

Output:

Integral((a + b*cot(c + d*x)**2)**(3/2), x)
 

Maxima [F]

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

integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*cot(d*x + c)^2 + a)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

int((a + b*cot(c + d*x)^2)^(3/2),x)
 

Output:

int((a + b*cot(c + d*x)^2)^(3/2), x)
 

Reduce [F]

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

int((a+b*cot(d*x+c)^2)^(3/2),x)
 

Output:

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