\(\int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx\) [105]

Optimal result

Integrand size = 27, antiderivative size = 132 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=-\frac {(i a-b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {(i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \cot (c+d x)}} \] Output:

-(I*a-b)*arctanh((a+b*cot(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d+(I* 
a+b)*arctanh((a+b*cot(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d-4*a*b/( 
a^2+b^2)/d/(a+b*cot(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.44 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\frac {b \left (\frac {\left (a^2-b^2+2 a \sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}+\frac {\left (-a^2+b^2+2 a \sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {-b^2} \sqrt {a+\sqrt {-b^2}}}-\frac {4 a}{\sqrt {a+b \cot (c+d x)}}\right )}{\left (a^2+b^2\right ) d} \] Input:

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

Output:

(b*(((a^2 - b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a 
- Sqrt[-b^2]]])/(Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) + ((-a^2 + b^2 + 2*a*Sqr 
t[-b^2])*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(Sqrt[-b^ 
2]*Sqrt[a + Sqrt[-b^2]]) - (4*a)/Sqrt[a + b*Cot[c + d*x]]))/((a^2 + b^2)*d 
)
 

Rubi [A] (warning: unable to verify)

Time = 0.67 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 4012, 25, 3042, 4022, 3042, 4020, 25, 73, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

\(\displaystyle \frac {\int -\frac {a^2-2 b \cot (c+d x) a-b^2}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {a^2-2 b \cot (c+d x) a-b^2}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {a^2+2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a-b^2}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {\frac {1}{2} (a-i b)^2 \int \frac {1-i \cot (c+d x)}{\sqrt {a+b \cot (c+d x)}}dx+\frac {1}{2} (a+i b)^2 \int \frac {i \cot (c+d x)+1}{\sqrt {a+b \cot (c+d x)}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {\frac {1}{2} (a-i b)^2 \int \frac {i \tan \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {1}{2} (a+i b)^2 \int \frac {1-i \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-b \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {\frac {i (a-i b)^2 \int -\frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}-\frac {i (a+i b)^2 \int -\frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}}{a^2+b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {\frac {i (a+i b)^2 \int \frac {1}{(1-i \cot (c+d x)) \sqrt {a+b \cot (c+d x)}}d(i \cot (c+d x))}{2 d}-\frac {i (a-i b)^2 \int \frac {1}{(i \cot (c+d x)+1) \sqrt {a+b \cot (c+d x)}}d(-i \cot (c+d x))}{2 d}}{a^2+b^2}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {-\frac {(a-i b)^2 \int \frac {1}{-\frac {i \cot ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}-\frac {(a+i b)^2 \int \frac {1}{\frac {i \cot ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \cot (c+d x)}}{b d}}{a^2+b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \cot (c+d x)}}-\frac {-\frac {(a-i b)^2 \arctan \left (\frac {\cot (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {(a+i b)^2 \arctan \left (\frac {\cot (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}}{a^2+b^2}\)

Input:

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

Output:

-((-(((a + I*b)^2*ArcTan[Cot[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) - 
 ((a - I*b)^2*ArcTan[Cot[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d))/(a^2 
+ b^2)) - (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 3042

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

rule 4012

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

rule 4020

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

rule 4022

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2284\) vs. \(2(112)=224\).

Time = 0.45 (sec) , antiderivative size = 2285, normalized size of antiderivative = 17.31

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2285\)
default	\(\text {Expression too large to display}\)	\(2285\)
parts	\(\text {Expression too large to display}\)	\(3675\)

Input:

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

Output:

-2/d*b^5/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot( 
d*x+c))^(1/2)-(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2) 
)+1/d*b^3/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot 
(d*x+c))^(1/2)-(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2 
))-2/d*b^5/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*co 
t(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 
2))+1/4/d*b^3/(a^2+b^2)^2*ln(b*cot(d*x+c)+a-(a+b*cot(d*x+c))^(1/2)*(2*(a^2 
+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/ 
d*b^3/(a^2+b^2)^2*ln(b*cot(d*x+c)+a+(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1 
/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/d*b^3/(a^2 
+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*cot(d*x+c))^(1/2) 
+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))-4*a*b/(a^2+ 
b^2)/d/(a+b*cot(d*x+c))^(1/2)+3/4/d*b^3/(a^2+b^2)^(5/2)*ln(b*cot(d*x+c)+a+ 
(a+b*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*( 
a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d*b^3/(a^2+b^2)^2/(2*(a^2+b^2)^(1/2)-2*a)^(1 
/2)*arctan((2*(a+b*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^ 
2+b^2)^(1/2)-2*a)^(1/2))*a-1/4/d/b/(a^2+b^2)^2*ln(b*cot(d*x+c)+a-(a+b*cot( 
d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^ 
(1/2)+2*a)^(1/2)*a^4-2/d*b^3/(a^2+b^2)^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arc 
tan((2*(a+b*cot(d*x+c))^(1/2)-(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2574 vs. \(2 (103) = 206\).

Time = 0.18 (sec) , antiderivative size = 2574, normalized size of antiderivative = 19.50 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(8*a*b*sqrt((b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 
 2*c))*sin(2*d*x + 2*c) - ((a^2*b + b^3)*d*cos(2*d*x + 2*c) + (a^3 + a*b^2 
)*d*sin(2*d*x + 2*c) + (a^2*b + b^3)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 
+ (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a 
^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2* 
b^4 + b^6)*d^2))*log((5*a^6*b - 5*a^4*b^3 - 9*a^2*b^5 + b^7)*sqrt((b*cos(2 
*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)/sin(2*d*x + 2*c)) + ((a^9 - 6*a^5*b^ 
4 - 8*a^3*b^6 - 3*a*b^8)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 
 - 20*a^2*b^8 + b^10)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a 
^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (15*a^6*b^2 - 35*a^4*b^4 + 13*a^2*b^6 
- b^8)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 
 + b^6)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b 
^10)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^ 
10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + ((a^2*b + 
b^3)*d*cos(2*d*x + 2*c) + (a^3 + a*b^2)*d*sin(2*d*x + 2*c) + (a^2*b + b^3) 
*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 
)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/( 
(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b 
^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log((5*a^6*b - 5...
 

Sympy [F]

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

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

Output:

-Integral(a/(a*sqrt(a + b*cot(c + d*x)) + b*sqrt(a + b*cot(c + d*x))*cot(c 
 + d*x)), x) - Integral(-b*cot(c + d*x)/(a*sqrt(a + b*cot(c + d*x)) + b*sq 
rt(a + b*cot(c + d*x))*cot(c + d*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 14.47 (sec) , antiderivative size = 5475, normalized size of antiderivative = 41.48 \[ \int \frac {-a+b \cot (c+d x)}{(a+b \cot (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

log(8*a*b^11*d^2 - (((((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - b^4*(16*a^6*d^ 
4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 12*a*b^4*d^2 + 
4*a^3*b^2*d^2)/(16*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^( 
1/2)*(64*a^6*b^7*d^4 - 96*a^2*b^11*d^4 - 64*a^4*b^9*d^4 - 32*b^13*d^4 + 96 
*a^8*b^5*d^4 + 32*a^10*b^3*d^4 + (a + b*cot(c + d*x))^(1/2)*((((24*a*b^4*d 
^2 - 8*a^3*b^2*d^2)^2/4 - b^4*(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 
48*a^4*b^2*d^4))^(1/2) - 12*a*b^4*d^2 + 4*a^3*b^2*d^2)/(16*(a^6*d^4 + b^6* 
d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10 
*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d 
^5)) + (a + b*cot(c + d*x))^(1/2)*(16*b^12*d^3 + 32*a^2*b^10*d^3 - 32*a^6* 
b^6*d^3 - 16*a^8*b^4*d^3))*((((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - b^4*(16 
*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 12*a*b^4 
*d^2 + 4*a^3*b^2*d^2)/(16*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d 
^4)))^(1/2) + 24*a^3*b^9*d^2 + 24*a^5*b^7*d^2 + 8*a^7*b^5*d^2)*((((24*a*b^ 
4*d^2 - 8*a^3*b^2*d^2)^2/4 - b^4*(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 
 + 48*a^4*b^2*d^4))^(1/2) - 12*a*b^4*d^2 + 4*a^3*b^2*d^2)/(16*(a^6*d^4 + b 
^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + log(8*a*b^11*d^2 - ((-(( 
(24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - b^4*(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2 
*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) + 12*a*b^4*d^2 - 4*a^3*b^2*d^2)/(16*(a^6 
*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(64*a^6*b^7*d^4...
 

Reduce [F]

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

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

Output:

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