3.1.99 \(\int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx\) [99]

3.1.99.1 Optimal result

Integrand size = 27, antiderivative size = 408 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \cot (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}+\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \left (a^2+b^2\right ) \log \left (a+\sqrt {a^2+b^2}+b \cot (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \cot (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \]

output

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

3.1.99.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.62 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.44 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\frac {(-a+b \cot (c+d x)) (a+b \cot (c+d x)) \left (3 i \sqrt {a-i b} \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a-i b}}\right )-3 i \sqrt {a+i b} \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \cot (c+d x)}}{\sqrt {a+i b}}\right )+2 b (a+b \cot (c+d x))^{3/2}\right ) \sin ^2(c+d x)}{-3 b^2 d \cos ^2(c+d x)+3 a^2 d \sin ^2(c+d x)} \]

input

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

output

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

3.1.99.3 Rubi [A] (warning: unable to verify)

Time = 0.71 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 4011, 27, 3042, 3966, 483, 1449, 1142, 25, 27, 1083, 219, 1103}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3966

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

\(\Big \downarrow \) 483

\(\displaystyle \frac {2 b \left (a^2+b^2\right ) \int \frac {b^2 \cot ^2(c+d x)}{b^4 \cot ^4(c+d x)-2 a b^2 \cot ^2(c+d x)+a^2+b^2}d\sqrt {a+b \cot (c+d x)}}{d}-\frac {2 b (a+b \cot (c+d x))^{3/2}}{3 d}\)

\(\Big \downarrow \) 1449

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1103

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

input

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

output

(-2*b*(a + b*Cot[c + d*x])^(3/2))/(3*d) + (2*b*(a^2 + b^2)*((-((Sqrt[a + S 
qrt[a^2 + b^2]]*ArcTanh[(-(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]) + 2*Sqrt[a + 
 b*Cot[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - Sqrt[a^2 
+ b^2]]) + Log[Sqrt[a^2 + b^2] + b^2*Cot[c + d*x]^2 - Sqrt[2]*Sqrt[a + Sqr 
t[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + 
b^2]]) - ((Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(Sqrt[2]*Sqrt[a + Sqrt[a^2 + 
b^2]] + 2*Sqrt[a + b*Cot[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/ 
Sqrt[a - Sqrt[a^2 + b^2]] + Log[Sqrt[a^2 + b^2] + b^2*Cot[c + d*x]^2 + Sqr 
t[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]]/2)/(2*Sqrt[2]*Sqr 
t[a + Sqrt[a^2 + b^2]])))/d
 

3.1.99.3.1 Defintions of rubi rules used

rule 25

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

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 483

Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d 
Subst[Int[x^2/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x 
] /; FreeQ[{a, b, c, d}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

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

rule 1449

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = 
Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r)   Int[x^(m - 1)/(q 
 - r*x + x^2), x], x] - Simp[1/(2*c*r)   Int[x^(m - 1)/(q + r*x + x^2), x], 
 x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 
3] && NegQ[b^2 - 4*a*c]
 

rule 3042

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

rule 3966

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Su 
bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c 
, d, n}, x] && NeQ[a^2 + b^2, 0]
 

rule 4011

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

3.1.99.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(333)=666\).

Time = 0.07 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.42

method	result	size
derivativedivides	\(-\frac {2 b \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}+\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{3} \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}-\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{3} \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}+\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}\)	\(986\)
default	\(-\frac {2 b \left (a +b \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}+\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\frac {b \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{3} \arctan \left (\frac {-2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \cot \left (d x +c \right )-\sqrt {a^{2}+b^{2}}-a \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}-\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}}{4 d b}-\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{3} \arctan \left (\frac {2 \sqrt {a +b \cot \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}}{4 d b}+\frac {b \ln \left (b \cot \left (d x +c \right )+a +\sqrt {a +b \cot \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d}\)	\(986\)
parts	\(\text {Expression too large to display}\)	\(1656\)

input

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

output

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

3.1.99.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (335) = 670\).

Time = 0.29 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.81 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

-1/6*(3*d*sqrt(-(a^5 + 2*a^3*b^2 + a*b^4 + d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 
+ 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4))/d^2)*log(d^3*sqrt(-(a^5 + 2*a^3*b^2 
+ a*b^4 + d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d 
^4))/d^2)*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) 
+ (a^6*b + 3*a^4*b^3 + 3*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)))*sin(2*d*x + 2*c) - 3*d*sqrt(-(a^5 + 2* 
a^3*b^2 + a*b^4 + d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + 
 b^10)/d^4))/d^2)*log(-d^3*sqrt(-(a^5 + 2*a^3*b^2 + a*b^4 + d^2*sqrt(-(a^8 
*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4))/d^2)*sqrt(-(a^8*b^2 
 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) + (a^6*b + 3*a^4*b^3 + 3 
*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)))*sin(2*d*x + 2*c) - 3*d*sqrt(-(a^5 + 2*a^3*b^2 + a*b^4 - d^2*sq 
rt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4))/d^2)*log(d^ 
3*sqrt(-(a^5 + 2*a^3*b^2 + a*b^4 - d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4* 
b^6 + 4*a^2*b^8 + b^10)/d^4))/d^2)*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 
+ 4*a^2*b^8 + b^10)/d^4) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sqrt((b*c 
os(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) + 3*d*sqrt(-(a^5 + 2*a^3*b^2 + a*b^4 - d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 
+ 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4))/d^2)*log(-d^3*sqrt(-(a^5 + 2*a^3*b^2 
 + a*b^4 - d^2*sqrt(-(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^1...
 

3.1.99.6 Sympy [F]

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

input

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

output

-Integral(a**2*sqrt(a + b*cot(c + d*x)), x) - Integral(-b**2*sqrt(a + b*co 
t(c + d*x))*cot(c + d*x)**2, x)
 

3.1.99.7 Maxima [F]

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

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

3.1.99.8 Giac [F]

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

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

3.1.99.9 Mupad [B] (verification not implemented)

Time = 25.17 (sec) , antiderivative size = 2529, normalized size of antiderivative = 6.20 \[ \int (-a+b \cot (c+d x)) (a+b \cot (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

log((((16*b^4*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^2))/d^2 - (1 
6*a*b^2*(((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) - 3*a*b^4*d^2 + a^3*b^2*d^2)/d^ 
4)^(1/2)*(a^2*b + b^3 + d*(((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) - 3*a*b^4*d^2 
 + a^3*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*(((-b^6*d^4*(3* 
a^2 - b^2)^2)^(1/2) - 3*a*b^4*d^2 + a^3*b^2*d^2)/d^4)^(1/2))/2 + (8*b^5*(a 
^2 - b^2)*(a^2 + b^2)^2)/d^3)*((6*a^2*b^8*d^4 - b^10*d^4 - 9*a^4*b^6*d^4)^ 
(1/2)/(4*d^4) - (3*a*b^4)/(4*d^2) + (a^3*b^2)/(4*d^2))^(1/2) - log((8*b^5* 
(a^2 - b^2)*(a^2 + b^2)^2)/d^3 - (((16*b^4*(a + b*cot(c + d*x))^(1/2)*(a^4 
 + b^4 - 6*a^2*b^2))/d^2 + (16*a*b^2*(-((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) + 
 3*a*b^4*d^2 - a^3*b^2*d^2)/d^4)^(1/2)*(a^2*b + b^3 - d*(-((-b^6*d^4*(3*a^ 
2 - b^2)^2)^(1/2) + 3*a*b^4*d^2 - a^3*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d 
*x))^(1/2)))/d)*(-((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) + 3*a*b^4*d^2 - a^3*b^ 
2*d^2)/d^4)^(1/2))/2)*(-((6*a^2*b^8*d^4 - b^10*d^4 - 9*a^4*b^6*d^4)^(1/2) 
+ 3*a*b^4*d^2 - a^3*b^2*d^2)/(4*d^4))^(1/2) - log((8*b^5*(a^2 - b^2)*(a^2 
+ b^2)^2)/d^3 - (((16*b^4*(a + b*cot(c + d*x))^(1/2)*(a^4 + b^4 - 6*a^2*b^ 
2))/d^2 + (16*a*b^2*(((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) - 3*a*b^4*d^2 + a^3 
*b^2*d^2)/d^4)^(1/2)*(a^2*b + b^3 - d*(((-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) - 
 3*a*b^4*d^2 + a^3*b^2*d^2)/d^4)^(1/2)*(a + b*cot(c + d*x))^(1/2)))/d)*((( 
-b^6*d^4*(3*a^2 - b^2)^2)^(1/2) - 3*a*b^4*d^2 + a^3*b^2*d^2)/d^4)^(1/2))/2 
)*(((6*a^2*b^8*d^4 - b^10*d^4 - 9*a^4*b^6*d^4)^(1/2) - 3*a*b^4*d^2 + a^...