\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [513]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 b d} \] Output:

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

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.17 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\frac {2 (a+b \tan (c+d x))^{5/2}}{b}+5 (i a+b) \left (\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-\sqrt {a+b \tan (c+d x)}\right )+5 i (a+i b) \left (-\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+b \tan (c+d x)}\right )}{5 d} \] Input:

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

Output:

((2*(a + b*Tan[c + d*x])^(5/2))/b + 5*(I*a + b)*(Sqrt[a - I*b]*ArcTanh[Sqr 
t[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - Sqrt[a + b*Tan[c + d*x]]) + (5*I)*( 
a + I*b)*(-(Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]) 
 + Sqrt[a + b*Tan[c + d*x]]))/(5*d)
 

Rubi [A] (warning: unable to verify)

Time = 0.69 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.85, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4026, 25, 3042, 3963, 3042, 4022, 3042, 4020, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3963

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 3963

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

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 4026

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(111)=222\).

Time = 0.19 (sec) , antiderivative size = 842, normalized size of antiderivative = 6.24

method	result	size
derivativedivides	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b d}-\frac {2 b \sqrt {a +b \tan \left (d x +c \right )}}{d}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}+\frac {b \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(842\)
default	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b d}-\frac {2 b \sqrt {a +b \tan \left (d x +c \right )}}{d}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}-\frac {b \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}+\frac {b \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(842\)

Input:

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

Output:

2/5*(a+b*tan(d*x+c))^(5/2)/b/d-2*b*(a+b*tan(d*x+c))^(1/2)/d-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))*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d/b*ln((a+b*ta 
n(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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^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))*(2*(a^2+b^2)^( 
1/2)+2*a)^(1/2)-1/d*b/(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^2+b^2)^(1/2)+2/d*b/(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+ 
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))*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^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))* 
(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+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))*(a^2+b^2)^(1/2)-2/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))*a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (105) = 210\).

Time = 0.11 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.97 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/10*(5*b*d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6 
)/d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) + ( 
a*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sqrt 
(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - 5 
*b*d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/ 
d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sq 
rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 
 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - 5*b*d*sqr 
t(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log 
(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(9*a 
^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 3*a*b^2 
 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) + 5*b*d*sqrt(-(a^3 
- 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)*log(-(3*a^4 
*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sqrt(-(9*a^4*b^2 - 
 6*a^2*b^4 + b^6)/d^4) - (3*a^2*b^2 - b^4)*d)*sqrt(-(a^3 - 3*a*b^2 - d^2*s 
qrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2)) - 4*(b^2*tan(d*x + c)^2 + 2 
*a*b*tan(d*x + c) + a^2 - 5*b^2)*sqrt(b*tan(d*x + c) + a))/(b*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[0,19,7]%%%}+%%%{8,[0,17,7]%%%}+%%%{28,[0,15,7]%%%}+ 
%%%{56,[0
 

Mupad [B] (verification not implemented)

Time = 7.27 (sec) , antiderivative size = 1141, normalized size of antiderivative = 8.45 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((b^6*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) 
 - a^3/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2)*32i)/((b^8*16i)/d - (16*a*b^7)/ 
d - (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d) + 
 (32*a*b^5*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2 
) - a^3/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((b^8*16i)/d - (16*a*b^7)/d - 
 (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (a 
^2*b^4*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) - 
a^3/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2)*96i)/((b^8*16i)/d - (16*a*b^7)/d - 
 (a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (9 
6*a^3*b^3*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2) 
 - a^3/(4*d^2) + (a^2*b*3i)/(4*d^2))^(1/2))/((b^8*16i)/d - (16*a*b^7)/d - 
(a^2*b^6*32i)/d + (32*a^3*b^5)/d - (a^4*b^4*48i)/d + (48*a^5*b^3)/d))*((3* 
a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2)*2i + atan((b^6*(a + b*tan( 
c + d*x))^(1/2)*((b^3*1i)/(4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2 
*b*3i)/(4*d^2))^(1/2)*32i)/((a^2*b^6*32i)/d - (16*a*b^7)/d - (b^8*16i)/d + 
 (32*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (32*a*b^5*(a + b*tan 
(c + d*x))^(1/2)*((b^3*1i)/(4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^ 
2*b*3i)/(4*d^2))^(1/2))/((a^2*b^6*32i)/d - (16*a*b^7)/d - (b^8*16i)/d + (3 
2*a^3*b^5)/d + (a^4*b^4*48i)/d + (48*a^5*b^3)/d) - (a^2*b^4*(a + b*tan(c + 
 d*x))^(1/2)*((b^3*1i)/(4*d^2) - a^3/(4*d^2) + (3*a*b^2)/(4*d^2) - (a^2...
 

Reduce [F]

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

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

Output:

int(sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**3,x)*b + int(sqrt(tan(c + d*x)* 
b + a)*tan(c + d*x)**2,x)*a