\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [524]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Output:

-5*a^(3/2)*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+I*(a-I*b)^(5/2)*arc 
tanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-I*(a+I*b)^(5/2)*arctanh((a+b* 
tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d-a^2*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.54 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {-5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-i a^2 \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 a \sqrt {a+i b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+i \sqrt {a+i b} b^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Input:

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

Output:

(-5*a^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + I*(a - I*b)^(5/2 
)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*a^2*Sqrt[a + I*b]*Ar 
cTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*a*Sqrt[a + I*b]*b*ArcTan 
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + I*Sqrt[a + I*b]*b^2*ArcTanh[Sq 
rt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - a^2*Cot[c + d*x]*Sqrt[a + b*Tan[c 
+ d*x]])/d
 

Rubi [A] (warning: unable to verify)

Time = 1.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 4048, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4048

\(\displaystyle \int \frac {\cot (c+d x) \left (5 b a^2-2 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {\cot (c+d x) \left (5 b a^2-2 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {5 b a^2-2 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (a^2-2 b^2\right ) \tan (c+d x)^2}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{2} \left (\int -\frac {2 \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx+5 a^2 b \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx\right )-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (5 a^2 b \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx-2 \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )-\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {1}{2} (a-i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {1}{2} (a-i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )\right )\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {i (a-i b)^3 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i (a+i b)^3 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {i (a+i b)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i (a-i b)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {(a-i b)^3 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {(a+i b)^3 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (5 a^2 b \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-2 \left (\frac {(a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (\frac {5 a^2 b \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}-2 \left (\frac {(a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (\frac {10 a^2 \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{d}-2 \left (\frac {(a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (-\frac {10 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-2 \left (\frac {(a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

Input:

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

Output:

(-2*(((a - I*b)^(5/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d + ((a + I*b)^( 
5/2)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d) - (10*a^(3/2)*b*ArcTanh[Sqrt[a 
 + b*Tan[c + d*x]]/Sqrt[a]])/d)/2 - (a^2*Cot[c + d*x]*Sqrt[a + b*Tan[c + d 
*x]])/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 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 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]
 

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 4117

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

rule 4136

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

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (119) = 238\).

Time = 0.20 (sec) , antiderivative size = 2372, normalized size of antiderivative = 15.71 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(5*a^(3/2)*b*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) 
 + 2*a)/tan(d*x + c))*tan(d*x + c) + d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + 
 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)/d^ 
4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) 
+ a) + ((a^2 - b^2)*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)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 
 10*a^3*b^2 + 5*a*b^4 + 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)/d^4))/d^2))*tan(d*x + c) - d*sqrt(-(a^5 - 10*a^3*b^2 
+ 5*a*b^4 + 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)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*ta 
n(d*x + c) + a) - ((a^2 - b^2)*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)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*s 
qrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 1 
10*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c) - d*sqrt(-(a^5 - 
10*a^3*b^2 + 5*a*b^4 - 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)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9 
)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*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)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 + 
a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - 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)/d^4))/d^2))*tan(d*x + c) + d...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 8.16 (sec) , antiderivative size = 3366, normalized size of antiderivative = 22.29 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d 
^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2 
)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4 
*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2* 
d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^ 
2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((1024*a^2*b^9*(a^2 + b^2))/d 
 - 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a 
*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x)) 
^(1/2)))/2 - (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(5*a^6 + 19*b^6 - 5*a^2* 
b^4 - 76*a^4*b^2))/d^2))/2 - (32*a*b^9*(b^8 - 23*a^8 - 100*a^2*b^6 + 44*a^ 
4*b^4 + 122*a^6*b^2))/d^3))/2 - (16*b^8*(a + b*tan(c + d*x))^(1/2)*(2*a^12 
 + 2*b^12 + 12*a^2*b^10 + 55*a^4*b^8 - 335*a^6*b^6 + 405*a^8*b^4 - 13*a^10 
*b^2))/d^4))/2 + (40*a^2*b^9*(a^2 + b^2)^3*(2*a^6 + 2*b^6 + 11*a^2*b^4 - 9 
*a^4*b^2))/d^5)*((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^ 
4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) - a^5/(4*d^2) - (5*a*b^4)/(4*d^2) + 
(5*a^3*b^2)/(2*d^2))^(1/2) - log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2 
)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4 
*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2* 
d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^ 
2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4...
 

Reduce [F]

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

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

Output:

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