\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx\) [709]

Optimal result

Integrand size = 21, antiderivative size = 193 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}-\frac {b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{2 \sqrt {-b^2} \left (a-\sqrt {-b^2}\right ) d (1+n)}+\frac {b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{2 \sqrt {-b^2} \left (a+\sqrt {-b^2}\right ) d (1+n)} \] Output:

(a+b*tan(d*x+c))^(1+n)/b/d/(1+n)-1/2*b*hypergeom([1, 1+n],[2+n],(a+b*tan(d 
*x+c))/(a-(-b^2)^(1/2)))*(a+b*tan(d*x+c))^(1+n)/(-b^2)^(1/2)/(a-(-b^2)^(1/ 
2))/d/(1+n)+1/2*b*hypergeom([1, 1+n],[2+n],(a+b*tan(d*x+c))/(a+(-b^2)^(1/2 
)))*(a+b*tan(d*x+c))^(1+n)/(-b^2)^(1/2)/(a+(-b^2)^(1/2))/d/(1+n)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.72 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {\left (i (a+i b) b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (2 a+2 i b-i b \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {a+b \tan (c+d x)}{a+i b}\right )\right )\right ) (a+b \tan (c+d x))^{1+n}}{2 b (-i a+b) (i a+b) d (1+n)} \] Input:

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

Output:

((I*(a + I*b)*b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a 
 - I*b)] + (a - I*b)*(2*a + (2*I)*b - I*b*Hypergeometric2F1[1, 1 + n, 2 + 
n, (a + b*Tan[c + d*x])/(a + I*b)]))*(a + b*Tan[c + d*x])^(1 + n))/(2*b*(( 
-I)*a + b)*(I*a + b)*d*(1 + n))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4026, 25, 3042, 3966, 485, 2009}

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.

\(\displaystyle \int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (c+d x)^2 (a+b \tan (c+d x))^ndx\)

\(\Big \downarrow \) 4026

\(\displaystyle \int -(a+b \tan (c+d x))^ndx+\frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}-\int (a+b \tan (c+d x))^ndx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}-\int (a+b \tan (c+d x))^ndx\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}-\frac {b \int \frac {(a+b \tan (c+d x))^n}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 485

\(\displaystyle \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}-\frac {b \int \left (\frac {\sqrt {-b^2} (a+b \tan (c+d x))^n}{2 b^2 \left (\sqrt {-b^2}-b \tan (c+d x)\right )}+\frac {\sqrt {-b^2} (a+b \tan (c+d x))^n}{2 b^2 \left (b \tan (c+d x)+\sqrt {-b^2}\right )}\right )d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)}-\frac {b \left (\frac {(a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} (n+1) \left (a-\sqrt {-b^2}\right )}-\frac {(a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} (n+1) \left (a+\sqrt {-b^2}\right )}\right )}{d}\)

Input:

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

Output:

(a + b*Tan[c + d*x])^(1 + n)/(b*d*(1 + n)) - (b*((Hypergeometric2F1[1, 1 + 
 n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 
+ n))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*(1 + n)) - (Hypergeometric2F1[1, 1 + 
n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + 
 n))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*(1 + n))))/d
 

Defintions of rubi rules used

rule 25

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

rule 485

Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand 
Integrand[(c + d*x)^n, 1/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, n}, x] & 
&  !IntegerQ[2*n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3966

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

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 [F]

\[\int \tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{n}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*tan(d*x + c) + a)^n*tan(d*x + c)^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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