\(\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) [816]

Optimal result

Integrand size = 23, antiderivative size = 190 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 a \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 b^2 (b+a \cot (c+d x))}{d \sqrt {\cot (c+d x)}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 2.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.99 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {-24 a^2 b+8 a^3 \cot (c+d x)-8 b \left (-3 a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 d \sqrt {\cot (c+d x)}} \] Input:

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

Output:

-1/4*(-24*a^2*b + 8*a^3*Cot[c + d*x] - 8*b*(-3*a^2 + b^2)*Hypergeometric2F 
1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a*(a^2 - 3*b^2)*Sqrt[Cot[c + d* 
x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[ 
Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[ 
1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(d*Sqrt[Cot[c + d*x]])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4113, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4048

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*a*(a^2 + b^2)*Sqrt[Cot[c + d*x]])/d + (2*b^2*(b + a*Cot[c + d*x]))/(d* 
Sqrt[Cot[c + d*x]]) + (2*(((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[ 
2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/S 
qrt[2]))/2 + ((a - b)*(a^2 + 4*a*b + b^2)*(-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]] + Co 
t[c + d*x]]/(2*Sqrt[2])))/2))/d
 

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 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 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

rule 4156

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(169)=338\).

Time = 0.22 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.87

method	result	size
derivativedivides	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, a^{2} b -\sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, b^{3}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{3}+6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{2} b +6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a \,b^{2}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, b^{3}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{3}+6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{2} b +6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a \,b^{2}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, b^{3}-\sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, a^{3}+3 \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, a \,b^{2}+8 b^{3} \tan \left (d x +c \right )-8 a^{3}\right )}{4 d}\)	\(545\)
default	\(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (3 \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, a^{2} b -\sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, b^{3}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{3}+6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{2} b +6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a \,b^{2}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, b^{3}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{3}+6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a^{2} b +6 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, a \,b^{2}-2 \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, b^{3}-\sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, a^{3}+3 \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, a \,b^{2}+8 b^{3} \tan \left (d x +c \right )-8 a^{3}\right )}{4 d}\)	\(545\)

Input:

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

Output:

1/4/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(3*tan(d*x+c)^(1/2)*ln(-(tan(d*x+c)+ 
2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/ 
2)*a^2*b-tan(d*x+c)^(1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^( 
1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*b^3-2*tan(d*x+c)^(1/2)*arctan 
(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a^3+6*tan(d*x+c)^(1/2)*arctan(1+2^(1/ 
2)*tan(d*x+c)^(1/2))*2^(1/2)*a^2*b+6*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan 
(d*x+c)^(1/2))*2^(1/2)*a*b^2-2*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c 
)^(1/2))*2^(1/2)*b^3-2*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2) 
)*2^(1/2)*a^3+6*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/ 
2)*a^2*b+6*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*a* 
b^2-2*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*b^3-tan 
(d*x+c)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1 
/2)*tan(d*x+c)^(1/2)+1))*2^(1/2)*a^3+3*tan(d*x+c)^(1/2)*ln(-(2^(1/2)*tan(d 
*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*2^(1/2) 
*a*b^2+8*b^3*tan(d*x+c)-8*a^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (169) = 338\).

Time = 0.09 (sec) , antiderivative size = 912, normalized size of antiderivative = 4.80 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(2*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b 
^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^ 
3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + 
b^6)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^ 
3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 
20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a^2*b 
^4 - b^6)) + 2*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 
3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 + 3*a^2*b - 3*a*b 
^2 - b^3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a 
*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 
- 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4 
*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 1 
5*a^2*b^4 - b^6)) + sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b 
^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*log(-a^3 - 3*a^2*b + 3*a*b^2 + b^3 + 
2*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6 
*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - (a^3 + 3*a^2*b - 3*a*b^2 - b^3)*ta 
n(d*x + c)) - sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3 
*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*log(-a^3 - 3*a^2*b + 3*a*b^2 + b^3 - 2*sqrt 
(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 
 + b^6)/d^2)*sqrt(tan(d*x + c)) - (a^3 + 3*a^2*b - 3*a*b^2 - b^3)*tan(d...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.15 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {8 \, b^{3} \sqrt {\tan \left (d x + c\right )} + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:

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

Output:

1/4*(8*b^3*sqrt(tan(d*x + c)) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)* 
arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^3 - 3* 
a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)) 
)) + sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)/sqrt(tan(d*x + c) 
) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqr 
t(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 8*a^3/sqrt(tan(d*x + c)))/ 
d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(cot(c + d*x))*a**3 - int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a** 
3*d + int(sqrt(cot(c + d*x))*cot(c + d*x)*tan(c + d*x)**3,x)*b**3*d + 3*in 
t(sqrt(cot(c + d*x))*cot(c + d*x)*tan(c + d*x)**2,x)*a*b**2*d + 3*int(sqrt 
(cot(c + d*x))*cot(c + d*x)*tan(c + d*x),x)*a**2*b*d)/d