\(\int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [525]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {\sqrt {a} \left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}-\frac {(a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \] Output:

1/4*a^(1/2)*(8*a^2-15*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d-(a-I* 
b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(5/2)*arc 
tanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d-9/4*a*b*cot(d*x+c)*(a+b*tan(d 
*x+c))^(1/2)/d-1/2*a^2*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 36.64 (sec) , antiderivative size = 22852, normalized size of antiderivative = 119.02 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Result too large to show} \] Input:

Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 1.74 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4048, 27, 3042, 4132, 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.

\(\displaystyle \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2}}{\tan (c+d x)^3}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {1}{2} \int \frac {\cot ^2(c+d x) \left (9 b a^2-4 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (3 a^2-4 b^2\right ) \tan ^2(c+d x)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {\cot ^2(c+d x) \left (9 b a^2-4 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (3 a^2-4 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {9 b a^2-4 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (3 a^2-4 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (9 b^2 \tan ^2(c+d x) a^2+\left (8 a^2-15 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (9 b^2 \tan ^2(c+d x) a^2+\left (8 a^2-15 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {9 b^2 \tan (c+d x)^2 a^2+\left (8 a^2-15 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {8 \left (a b \left (3 a^2-b^2\right )-a^2 \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \left (8 a^2-15 b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {8 \int \frac {a b \left (3 a^2-b^2\right )-a^2 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a^2 \left (8 a^2-15 b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \int \frac {a b \left (3 a^2-b^2\right )-a^2 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} a (-b+i a)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a (b+i a)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} a (-b+i a)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a (b+i a)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (-\frac {i a (-b+i a)^3 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i a (b+i a)^3 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {i a (-b+i a)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {i a (b+i a)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {a (-b+i a)^3 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {a (b+i a)^3 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {a (-b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a (b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{2 a}\right )\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {a^2 \left (8 a^2-15 b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+8 \left (\frac {a (-b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a (b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{2 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {2 a^2 \left (8 a^2-15 b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+8 \left (\frac {a (-b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a (b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{2 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {9 a b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {-\frac {2 a^{3/2} \left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+8 \left (\frac {a (-b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a (b+i a)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{2 a}\right )\)

Input:

Int[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2),x]
 

Output:

-1/2*(a^2*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d + (-1/2*(8*(-((a*(I*a 
 + b)^3*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + (a*(I*a - 
 b)^3*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (2*a^(3/2)* 
(8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d)/a - (9*a*b* 
Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d)/4
 

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 4132

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[e + f*x]^2, 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] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

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 [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 26.63 (sec) , antiderivative size = 1541429, normalized size of antiderivative = 8028.28

\[\text {output too large to display}\]

Input:

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (156) = 312\).

Time = 0.24 (sec) , antiderivative size = 2395, normalized size of antiderivative = 12.47 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

[1/8*(4*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 - 14*a^4*b 
^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*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) - (5*a^6 - 15*a^4* 
b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(2 
5*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)^2 - 4*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 - 
14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) - (2*a*d^3*sqrt(-(2 
5*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - (5*a^6 - 
 15*a^4*b^2 + 11*a^2*b^4 - 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)^2 - 4*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 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*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) + 
(5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - 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)^2 + 4*d*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*s 
qrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))...
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(d*x+c)^3*(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 = 8.60 (sec) , antiderivative size = 4304, normalized size of antiderivative = 22.42 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

int(cot(c + d*x)^3*(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)*(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) - (64*a*b^8*(13*b^4 - 
6*a^4 + 7*a^2*b^2))/d))/2 - (4*a*b^8*(a + b*tan(c + d*x))^(1/2)*(144*a^6 + 
 304*b^6 + 145*a^2*b^4 - 1056*a^4*b^2))/d^2))/2 - (6*a^2*b^8*(16*a^8 - 309 
*b^8 + 576*a^2*b^6 + 485*a^4*b^4 - 384*a^6*b^2))/d^3))/2 + (b^8*(a + b*tan 
(c + d*x))^(1/2)*(96*a^12 + 32*b^12 - 33*a^2*b^10 + 4095*a^4*b^8 - 6399*a^ 
6*b^6 + 5265*a^8*b^4 - 1008*a^10*b^2))/d^4))/2 - (a*b^10*(a^2 + b^2)^3*(50 
4*a^6 - 120*b^6 + 379*a^2*b^4 - 1113*a^4*b^2))/(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^...
 

Reduce [F]

\[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int \cot \left (d x +c \right )^{3} \left (a +\tan \left (d x +c \right ) b \right )^{\frac {5}{2}}d x \] Input:

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(5/2),x)
 

Output:

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(5/2),x)