3.4.23 \(\int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) [323]

3.4.23.1 Optimal result

Integrand size = 33, antiderivative size = 219 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A+A b^2-4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d}-\frac {\sqrt {a-i b} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a d}-\frac {A \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

output

1/4*(8*A*a^2+A*b^2-4*B*a*b)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2 
)/d-(A-I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))*(a-I*b)^(1/2)/d- 
(A+I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))*(a+I*b)^(1/2)/d-1/4* 
(A*b+4*B*a)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a/d-1/2*A*cot(d*x+c)^2*(a+b* 
tan(d*x+c))^(1/2)/d
 

3.4.23.2 Mathematica [A] (verified)

Time = 4.92 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.24 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\frac {\left (8 a^2 A+A b^2-4 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\frac {4 \left (-a A b+A b \sqrt {-b^2}+b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}-\frac {4 \left (a A b+A b \sqrt {-b^2}-b^2 B+a \sqrt {-b^2} B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}-\frac {b \cot (c+d x) (A b+4 a B+2 a A \cot (c+d x)) \sqrt {a+b \tan (c+d x)}}{a}}{b}}{4 d} \]

input

Integrate[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]
 

output

(((8*a^2*A + A*b^2 - 4*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a 
^(3/2) + ((4*(-(a*A*b) + A*b*Sqrt[-b^2] + b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[ 
Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - (4* 
(a*A*b + A*b*Sqrt[-b^2] - b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c 
 + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] - (b*Cot[c + d*x]*(A* 
b + 4*a*B + 2*a*A*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/a)/b)/(4*d)
 

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

Time = 1.75 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4091, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4091

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

input

Int[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]
 

output

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

3.4.23.3.1 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 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 4091

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

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]
 

3.4.23.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1144\) vs. \(2(185)=370\).

Time = 0.25 (sec) , antiderivative size = 1145, normalized size of antiderivative = 5.23

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1145\)
default	\(\text {Expression too large to display}\)	\(1145\)

input

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x,method=_RETURNV 
ERBOSE)
 

output

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

3.4.23.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1338 vs. \(2 (179) = 358\).

Time = 3.17 (sec) , antiderivative size = 2691, normalized size of antiderivative = 12.29 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

input

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorith 
m="fricas")
 

output

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

3.4.23.6 Sympy [F]

\[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx \]

input

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**(1/2)*(A+B*tan(d*x+c)),x)
 

output

Integral((A + B*tan(c + d*x))*sqrt(a + b*tan(c + d*x))*cot(c + d*x)**3, x)
 

3.4.23.7 Maxima [F]

\[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{3} \,d x } \]

input

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorith 
m="maxima")
 

output

integrate((B*tan(d*x + c) + A)*sqrt(b*tan(d*x + c) + a)*cot(d*x + c)^3, x)
 

3.4.23.8 Giac [F(-1)]

Timed out. \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

input

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorith 
m="giac")
 

output

Timed out
 

3.4.23.9 Mupad [B] (verification not implemented)

Time = 9.46 (sec) , antiderivative size = 14195, normalized size of antiderivative = 64.82 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

input

int(cot(c + d*x)^3*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(1/2),x)
 

output

(atan(((((((2*A^3*b^14*d^2 + 2*A^3*a^2*b^12*d^2 - 96*A^3*a^4*b^10*d^2 - 96 
*A^3*a^6*b^8*d^2 - 160*B^3*a^3*b^11*d^2 - 160*B^3*a^5*b^9*d^2 + 48*A^2*B*a 
*b^13*d^2 - 192*A*B^2*a^2*b^12*d^2 + 96*A*B^2*a^4*b^10*d^2 + 288*A*B^2*a^6 
*b^8*d^2 + 528*A^2*B*a^3*b^11*d^2 + 480*A^2*B*a^5*b^9*d^2)/(8*a^2*d^5) + ( 
((((64*A*a*b^12*d^4 + 448*A*a^3*b^10*d^4 + 384*A*a^5*b^8*d^4 - 256*B*a^2*b 
^11*d^4 - 256*B*a^4*b^9*d^4)/(8*a^2*d^5) - ((512*a^2*b^10*d^4 + 768*a^4*b^ 
8*d^4)*(a + b*tan(c + d*x))^(1/2)*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b 
^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(64*a^5*d^5))*( 
64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b 
- 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(128*B^2*a 
^3*b^10*d^2 - 576*A^2*a^5*b^8*d^2 - 256*A^2*a^3*b^10*d^2 + 320*B^2*a^5*b^8 
*d^2 - 4*A^2*a*b^12*d^2 + 544*A*B*a^2*b^11*d^2 + 1024*A*B*a^4*b^9*d^2))/(8 
*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4 + 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 6 
4*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8*a^3*d))*(64*A^2*a^7 + A^2*a^3*b^4 + 
 16*A^2*a^5*b^2 + 16*B^2*a^5*b^2 - 64*A*B*a^6*b - 8*A*B*a^4*b^3)^(1/2))/(8 
*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^14 - A^4*b^14 + 17*A^4*a^ 
2*b^12 + 16*A^4*a^4*b^10 + 96*A^4*a^6*b^8 + 48*B^4*a^2*b^12 + 48*B^4*a^4*b 
^10 + 32*B^4*a^6*b^8 + 95*A^2*B^2*a^2*b^12 + 448*A^2*B^2*a^4*b^10 - 8*A*B^ 
3*a*b^13 + 4*A^3*B*a*b^13 - 120*A*B^3*a^3*b^11 + 64*A*B^3*a^5*b^9 - 8*A^3* 
B*a^3*b^11 - 320*A^3*B*a^5*b^9))/(8*a^2*d^4))*(64*A^2*a^7 + A^2*a^3*b^4...