\(\int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [512]

Optimal result

Integrand size = 23, antiderivative size = 181 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}-\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {4 a (a+b \tan (c+d x))^{5/2}}{35 b^2 d}+\frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d} \] 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-4/35*a*(a+b*tan(d*x+c))^(5/2)/b^2/d+2/7* 
tan(d*x+c)*(a+b*tan(d*x+c))^(5/2)/b/d
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \sqrt {a+b \tan (c+d x)} \left (-2 a \left (3 a^2+70 b^2\right )+b \left (3 a^2-35 b^2\right ) \tan (c+d x)+24 a b^2 \tan ^2(c+d x)+15 b^3 \tan ^3(c+d x)\right )}{105 b^2 d} \] Input:

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

Output:

((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a 
+ I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*Sqrt[ 
a + b*Tan[c + d*x]]*(-2*a*(3*a^2 + 70*b^2) + b*(3*a^2 - 35*b^2)*Tan[c + d* 
x] + 24*a*b^2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/(105*b^2*d)
 

Rubi [A] (warning: unable to verify)

Time = 1.16 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4049, 27, 3042, 4113, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4049

\(\displaystyle \frac {2 \int -\frac {1}{2} (a+b \tan (c+d x))^{3/2} \left (2 a \tan ^2(c+d x)+7 b \tan (c+d x)+2 a\right )dx}{7 b}+\frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\int (a+b \tan (c+d x))^{3/2} \left (2 a \tan ^2(c+d x)+7 b \tan (c+d x)+2 a\right )dx}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\int (a+b \tan (c+d x))^{3/2} \left (2 a \tan (c+d x)^2+7 b \tan (c+d x)+2 a\right )dx}{7 b}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\int 7 b \tan (c+d x) (a+b \tan (c+d x))^{3/2}dx+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \int \tan (c+d x) (a+b \tan (c+d x))^{3/2}dx+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \int \tan (c+d x) (a+b \tan (c+d x))^{3/2}dx+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \left (\int (a \tan (c+d x)-b) \sqrt {a+b \tan (c+d x)}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}\right )+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \left (\int (a \tan (c+d x)-b) \sqrt {a+b \tan (c+d x)}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}\right )+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \left (\int \frac {\left (a^2-b^2\right ) \tan (c+d x)-2 a b}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {7 b \left (\int \frac {\left (a^2-b^2\right ) \tan (c+d x)-2 a b}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )+\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}}{7 b}\)

\(\Big \downarrow \) 4022

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (-\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (-\frac {1}{2} i (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} i (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (\frac {(a-i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {(a+i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (-\frac {(a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {(a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (-\frac {i (a-i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {i (a+i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac {\frac {4 a (a+b \tan (c+d x))^{5/2}}{5 b d}+7 b \left (-\frac {i (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}+\frac {2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 a \sqrt {a+b \tan (c+d x)}}{d}\right )}{7 b}\)

Input:

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

Output:

(2*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2))/(7*b*d) - ((4*a*(a + b*Tan[c + 
 d*x])^(5/2))/(5*b*d) + 7*b*(((-I)*(a - I*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqr 
t[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)) 
)/(7*b)
 

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 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]
 

rule 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4113

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(862\) vs. \(2(151)=302\).

Time = 0.20 (sec) , antiderivative size = 863, normalized size of antiderivative = 4.77

method	result	size
derivativedivides	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d \,b^{2}}-\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b^{2} d}-\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 ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{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 )}{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 ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{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 )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(863\)
default	\(\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d \,b^{2}}-\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b^{2} d}-\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 ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b^{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 )}{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 ) a^{2}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b^{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 )}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(863\)

Input:

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

Output:

2/7/d/b^2*(a+b*tan(d*x+c))^(7/2)-2/5*a*(a+b*tan(d*x+c))^(5/2)/b^2/d-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*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^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^2-1/d*b^2/(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))-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)*arcta 
n((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+1/d*b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*ta 
n(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)^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (147) = 294\).

Time = 0.10 (sec) , antiderivative size = 817, normalized size of antiderivative = 4.51 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/210*(105*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)*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)) - 105*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)*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)) - 105*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)*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)) + 105*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)*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*(15*b^3*tan(d*x + c)^3 + 24*a*b^2*tan(d 
*x + c)^2 - 6*a^3 - 140*a*b^2 + (3*a^2*b - 35*b^3)*tan(d*x + c))*sqrt(b*ta 
n(d*x + c) + a))/(b^2*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(tan(d*x+c)^3*(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)^3, x)
 

Giac [F(-2)]

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

integrate(tan(d*x+c)^3*(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,[0,24,9]%%%}+%%%{10,[0,22,9]%%%}+%%%{45,[0,20,9]%%%} 
+%%%{120,
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 19.16 (sec) , antiderivative size = 1229, normalized size of antiderivative = 6.79 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \] Input:

int(tan(c + d*x)^3*(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)/((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*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))/((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^2*b*3i)/(4*d^2))^(1/2)*96i)/((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) + (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))/((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))*(-(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)/((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) + (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))/((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^...
 

Reduce [F]

\[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\left (\int \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{4}d x \right ) b +\left (\int \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{3}d x \right ) a \] Input:

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

Output:

int(sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**4,x)*b + int(sqrt(tan(c + d*x)* 
b + a)*tan(c + d*x)**3,x)*a