3.6.74 \(\int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx\) [574]

3.6.74.1 Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {(b (A-B)+a (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 (A b+a B) \sqrt {\cot (c+d x)}}{d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {(a (A-B)-b (A+B)) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]

output

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

3.6.74.2 Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.86 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {\sqrt {\cot (c+d x)} \left (6 \sqrt {2} (b (A-B)+a (A+B)) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+3 \sqrt {2} (a (A-B)-b (A+B)) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )-\frac {8 a A}{\tan ^{\frac {3}{2}}(c+d x)}-\frac {24 (A b+a B)}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{12 d} \]

input

Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]
 

output

(Sqrt[Cot[c + d*x]]*(6*Sqrt[2]*(b*(A - B) + a*(A + B))*(ArcTan[1 - Sqrt[2] 
*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 3*Sqrt[2] 
*(a*(A - B) - b*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x 
]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) - (8*a*A)/Tan[c + 
 d*x]^(3/2) - (24*(A*b + a*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12 
*d)
 

3.6.74.3 Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.90, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4064, 3042, 4075, 3042, 4011, 3042, 4017, 1482, 1476, 1082, 217, 1479, 25, 27, 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.

\(\displaystyle \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (c+d x)^{5/2} (a+b \tan (c+d x)) (A+B \tan (c+d x))dx\)

\(\Big \downarrow \) 4064

\(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+b) (A \cot (c+d x)+B)dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4075

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

input

Int[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]
 

output

(-2*(A*b + a*B)*Sqrt[Cot[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2))/(3*d) + 
 (2*(((b*(A - B) + a*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sq 
rt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 - ((a*(A - B) 
- b*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt 
[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2)) 
/d
 

3.6.74.3.1 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 217

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

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

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]
 

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 4064

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp 
[g^(m + n)   Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c 
*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !Integer 
Q[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4075

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

3.6.74.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs. \(2(195)=390\).

Time = 0.47 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +3 A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b -3 B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a -6 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a -6 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +3 B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +24 A \tan \left (d x +c \right ) b +24 B \tan \left (d x +c \right ) a +8 A a \right )}{12 d}\)	\(535\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (3 A \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +6 A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +3 A \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b -3 B \sqrt {2}\, \ln \left (-\frac {1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a -6 B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +6 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a -6 B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} b +3 B \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+\tan \left (d x +c \right )}\right ) \tan \left (d x +c \right )^{\frac {3}{2}} a +24 A \tan \left (d x +c \right ) b +24 B \tan \left (d x +c \right ) a +8 A a \right )}{12 d}\)	\(535\)

input

int(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x,method=_RETURNVER 
BOSE)
 

output

-1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(3*A*2^(1/2)*ln(-(1+2^(1/2)*tan(d* 
x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*tan(d*x+c) 
^(3/2)*a+6*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*a 
+6*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*b+6*A*2^( 
1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*a+6*A*2^(1/2)*ar 
ctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*b+3*A*2^(1/2)*ln(-(2^(1 
/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)) 
)*tan(d*x+c)^(3/2)*b-3*B*2^(1/2)*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c 
))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*tan(d*x+c)^(3/2)*b+6*B*2^(1/2) 
*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*a-6*B*2^(1/2)*arctan( 
1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*b+6*B*2^(1/2)*arctan(-1+2^(1/ 
2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)*a-6*B*2^(1/2)*arctan(-1+2^(1/2)*tan( 
d*x+c)^(1/2))*tan(d*x+c)^(3/2)*b+3*B*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2) 
-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*tan(d*x+c)^(3/2)*a 
+24*A*tan(d*x+c)*b+24*B*tan(d*x+c)*a+8*A*a)
 

3.6.74.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2267 vs. \(2 (195) = 390\).

Time = 0.70 (sec) , antiderivative size = 2267, normalized size of antiderivative = 9.90 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

input

integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm= 
"fricas")
 

output

-1/6*(3*d*sqrt(-(2*A*B*a^2 - 2*A*B*b^2 + 2*(A^2 - B^2)*a*b + d^2*sqrt(-((A 
^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 
+ B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^ 
4))/d^2)*log(((B*a + A*b)*d^3*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3* 
B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)* 
a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + ((A^3 - A*B^2)*a^3 - (5*A^2*B 
- B^3)*a^2*b - (A^3 - 5*A*B^2)*a*b^2 + (A^2*B - B^3)*b^3)*d)*sqrt(-(2*A*B* 
a^2 - 2*A*B*b^2 + 2*(A^2 - B^2)*a*b + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a 
^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3 
*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))/d^2) + ((A^4 - B^4) 
*a^4 - 4*(A^3*B + A*B^3)*a^3*b - 4*(A^3*B + A*B^3)*a*b^3 - (A^4 - B^4)*b^4 
)*sqrt(tan(d*x + c)))*tan(d*x + c) - 3*d*sqrt(-(2*A*B*a^2 - 2*A*B*b^2 + 2* 
(A^2 - B^2)*a*b + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^ 
3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + 
(A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))/d^2)*log(-((B*a + A*b)*d^3*sqrt(-((A^4 
- 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B 
^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) 
+ ((A^3 - A*B^2)*a^3 - (5*A^2*B - B^3)*a^2*b - (A^3 - 5*A*B^2)*a*b^2 + (A^ 
2*B - B^3)*b^3)*d)*sqrt(-(2*A*B*a^2 - 2*A*B*b^2 + 2*(A^2 - B^2)*a*b + d^2* 
sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 -...
 

3.6.74.6 Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

input

integrate(cot(d*x+c)**(5/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x)
 

output

Timed out
 

3.6.74.7 Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.86 \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {6 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, A a}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {24 \, {\left (B a + A b\right )}}{\sqrt {\tan \left (d x + c\right )}}}{12 \, d} \]

input

integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm= 
"maxima")
 

output

1/12*(6*sqrt(2)*((A + B)*a + (A - B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sq 
rt(tan(d*x + c)))) + 6*sqrt(2)*((A + B)*a + (A - B)*b)*arctan(-1/2*sqrt(2) 
*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 3*sqrt(2)*((A - B)*a - (A + B)*b)*log 
(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 3*sqrt(2)*((A - B)*a - 
 (A + B)*b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 8*A*a/ 
tan(d*x + c)^(3/2) - 24*(B*a + A*b)/sqrt(tan(d*x + c)))/d
 

3.6.74.8 Giac [F]

\[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {5}{2}} \,d x } \]

input

integrate(cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm= 
"giac")
 

output

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)*cot(d*x + c)^(5/2), x)
 

3.6.74.9 Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right ) \,d x \]

input

int(cot(c + d*x)^(5/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x)),x)
 

output

int(cot(c + d*x)^(5/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x)), x)