3.1.45 \(\int \tan ^2(c+d x) (b \tan (c+d x))^n (A+B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [45]

3.1.45.1 Optimal result

Integrand size = 39, antiderivative size = 132 \[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {(A-C) \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2},\frac {5+n}{2},-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {B \operatorname {Hypergeometric2F1}\left (1,\frac {4+n}{2},\frac {6+n}{2},-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{4+n}}{b^4 d (4+n)} \]

output

C*(b*tan(d*x+c))^(3+n)/b^3/d/(3+n)+(A-C)*hypergeom([1, 3/2+1/2*n],[5/2+1/2 
*n],-tan(d*x+c)^2)*(b*tan(d*x+c))^(3+n)/b^3/d/(3+n)+B*hypergeom([1, 2+1/2* 
n],[3+1/2*n],-tan(d*x+c)^2)*(b*tan(d*x+c))^(4+n)/b^4/d/(4+n)
 

3.1.45.2 Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\tan ^3(c+d x) (b \tan (c+d x))^n \left (C (4+n)+(A-C) (4+n) \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{2},\frac {5+n}{2},-\tan ^2(c+d x)\right )+B (3+n) \operatorname {Hypergeometric2F1}\left (1,\frac {4+n}{2},\frac {6+n}{2},-\tan ^2(c+d x)\right ) \tan (c+d x)\right )}{d (3+n) (4+n)} \]

input

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

output

(Tan[c + d*x]^3*(b*Tan[c + d*x])^n*(C*(4 + n) + (A - C)*(4 + n)*Hypergeome 
tric2F1[1, (3 + n)/2, (5 + n)/2, -Tan[c + d*x]^2] + B*(3 + n)*Hypergeometr 
ic2F1[1, (4 + n)/2, (6 + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]))/(d*(3 + n)* 
(4 + n))
 

3.1.45.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2030, 3042, 4113, 3042, 4021, 3042, 3957, 278}

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.

\(\displaystyle \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 2030

\(\displaystyle \frac {\int (b \tan (c+d x))^{n+2} \left (C \tan ^2(c+d x)+B \tan (c+d x)+A\right )dx}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (b \tan (c+d x))^{n+2} \left (C \tan (c+d x)^2+B \tan (c+d x)+A\right )dx}{b^2}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {\int (b \tan (c+d x))^{n+2} (A-C+B \tan (c+d x))dx+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (b \tan (c+d x))^{n+2} (A-C+B \tan (c+d x))dx+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

\(\Big \downarrow \) 4021

\(\displaystyle \frac {(A-C) \int (b \tan (c+d x))^{n+2}dx+\frac {B \int (b \tan (c+d x))^{n+3}dx}{b}+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(A-C) \int (b \tan (c+d x))^{n+2}dx+\frac {B \int (b \tan (c+d x))^{n+3}dx}{b}+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\frac {b (A-C) \int \frac {(b \tan (c+d x))^{n+2}}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}+\frac {B \int \frac {(b \tan (c+d x))^{n+3}}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {(A-C) (b \tan (c+d x))^{n+3} \operatorname {Hypergeometric2F1}\left (1,\frac {n+3}{2},\frac {n+5}{2},-\tan ^2(c+d x)\right )}{b d (n+3)}+\frac {B (b \tan (c+d x))^{n+4} \operatorname {Hypergeometric2F1}\left (1,\frac {n+4}{2},\frac {n+6}{2},-\tan ^2(c+d x)\right )}{b^2 d (n+4)}+\frac {C (b \tan (c+d x))^{n+3}}{b d (n+3)}}{b^2}\)

input

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

output

((C*(b*Tan[c + d*x])^(3 + n))/(b*d*(3 + n)) + ((A - C)*Hypergeometric2F1[1 
, (3 + n)/2, (5 + n)/2, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(3 + n))/(b*d*(3 
 + n)) + (B*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Tan[c + d*x]^2]*(b 
*Tan[c + d*x])^(4 + n))/(b^2*d*(4 + n)))/b^2
 

3.1.45.3.1 Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 2030

Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m   Int[(b*v) 
^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
 

rule 3042

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4021

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

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]
 

3.1.45.4 Maple [F]

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

input

int(tan(d*x+c)^2*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)
 

output

int(tan(d*x+c)^2*(b*tan(d*x+c))^n*(A+B*tan(d*x+c)+C*tan(d*x+c)^2),x)
 

3.1.45.5 Fricas [F]

\[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{2} \,d x } \]

input

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

output

integral((C*tan(d*x + c)^4 + B*tan(d*x + c)^3 + A*tan(d*x + c)^2)*(b*tan(d 
*x + c))^n, x)
 

3.1.45.6 Sympy [F]

\[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int \left (b \tan {\left (c + d x \right )}\right )^{n} \left (A + B \tan {\left (c + d x \right )} + C \tan ^{2}{\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}\, dx \]

input

integrate(tan(d*x+c)**2*(b*tan(d*x+c))**n*(A+B*tan(d*x+c)+C*tan(d*x+c)**2) 
,x)
 

output

Integral((b*tan(c + d*x))**n*(A + B*tan(c + d*x) + C*tan(c + d*x)**2)*tan( 
c + d*x)**2, x)
 

3.1.45.7 Maxima [F]

\[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{2} \,d x } \]

input

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

output

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d 
*x + c)^2, x)
 

3.1.45.8 Giac [F]

\[ \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\int { {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{2} \,d x } \]

input

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

output

integrate((C*tan(d*x + c)^2 + B*tan(d*x + c) + A)*(b*tan(d*x + c))^n*tan(d 
*x + c)^2, x)
 

3.1.45.9 Mupad [F(-1)]

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

input

int(tan(c + d*x)^2*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x) 
^2),x)
 

output

int(tan(c + d*x)^2*(b*tan(c + d*x))^n*(A + B*tan(c + d*x) + C*tan(c + d*x) 
^2), x)