\(\int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [514]

Optimal result

Integrand size = 21, antiderivative size = 128 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {-3 (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-3 (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 \sqrt {a+b \tan (c+d x)} (4 a+b \tan (c+d x))}{3 d} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.74 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4011, 3042, 4011, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

\(\displaystyle \frac {(a-i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {(a+i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {(a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {i (a-i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a+i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

((-I)*(a - I*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d + (I*(a + I*b) 
^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d + (2*a*Sqrt[a + b*Tan[c + d*x 
]])/d + (2*(a + b*Tan[c + d*x])^(3/2))/(3*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 4011

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(106)=212\).

Time = 0.18 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.49

method	result	size
derivativedivides	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {2 a \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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{2 d}+\frac {\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}}\, a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\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 ) \left (a^{2}+b^{2}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 \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^{2}}{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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\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 d}-\frac {\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}}\, a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) \left (a^{2}+b^{2}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 \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^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(831\)
default	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {2 a \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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{2 d}+\frac {\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}}\, a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\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 ) \left (a^{2}+b^{2}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {2 \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^{2}}{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 {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d}-\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 d}-\frac {\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}}\, a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\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 ) \left (a^{2}+b^{2}\right )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {2 \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^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(831\)

Input:

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

Output:

2/3*(a+b*tan(d*x+c))^(3/2)/d+2*a*(a+b*tan(d*x+c))^(1/2)/d-1/4/d*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+b^2)^(1/2)+1/2/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))*(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^2+b^2)^(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^2+b^2)-2/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 
^2+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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)-1/2/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))*(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^2+b^2)^(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^2+b^2)+2/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arc 
tan((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
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 756 vs. \(2 (102) = 204\).

Time = 0.09 (sec) , antiderivative size = 756, normalized size of antiderivative = 5.91 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(3*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sq 
rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*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)) - 3*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3*sqrt(-(9*a^4*b^2 - 6 
*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^2)*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)) - 3*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 + 2*a^2*b^2 - b 
^4)*sqrt(b*tan(d*x + c) + a) + (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^ 
4) + (3*a^3 - a*b^2)*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)) + 3*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x 
 + c) + a) - (d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^3 - a*b^ 
2)*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)) - 4*(b*tan(d*x + c) + 4*a)*sqrt(b*tan(d*x + c) + a))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more 
details)Is
 

Giac [F(-2)]

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

integrate(tan(d*x+c)*(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,[0,14,5]%%%}+%%%{6,[0,12,5]%%%}+%%%{15,[0,10,5]%%%}+ 
%%%{20,[0
 

Mupad [B] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 1112, normalized size of antiderivative = 8.69 \[ \int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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