\(\int \tan ^5(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [361]

Optimal result

Integrand size = 23, antiderivative size = 129 \[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 b^2 f (1+p)}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 (a-b) f (1+p)}+\frac {\left (a+b \tan ^2(e+f x)\right )^{2+p}}{2 b^2 f (2+p)} \] Output:

-1/2*(a+b)*(a+b*tan(f*x+e)^2)^(p+1)/b^2/f/(p+1)-1/2*hypergeom([1, p+1],[2+ 
p],(a+b*tan(f*x+e)^2)/(a-b))*(a+b*tan(f*x+e)^2)^(p+1)/(a-b)/f/(p+1)+1/2*(a 
+b*tan(f*x+e)^2)^(2+p)/b^2/f/(2+p)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\left (a+b \tan ^2(e+f x)\right )^{1+p} \left (b^2 (2+p) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (a+b (2+p)-b (1+p) \tan ^2(e+f x)\right )\right )}{2 b^2 (-a+b) f (1+p) (2+p)} \] Input:

Integrate[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]
 

Output:

((a + b*Tan[e + f*x]^2)^(1 + p)*(b^2*(2 + p)*Hypergeometric2F1[1, 1 + p, 2 
 + p, (a + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a + b*(2 + p) - b*(1 + p) 
*Tan[e + f*x]^2)))/(2*b^2*(-a + b)*f*(1 + p)*(2 + p))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4153, 354, 99, 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.

\(\displaystyle \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^5 \left (a+b \tan (e+f x)^2\right )^pdx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {\int \frac {\tan ^5(e+f x) \left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\tan ^4(e+f x) \left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}d\tan ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (\frac {(-a-b) \left (b \tan ^2(e+f x)+a\right )^p}{b}+\frac {\left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}+\frac {\left (b \tan ^2(e+f x)+a\right )^{p+1}}{b}\right )d\tan ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {(a+b) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{b^2 (p+1)}+\frac {\left (a+b \tan ^2(e+f x)\right )^{p+2}}{b^2 (p+2)}-\frac {\left (a+b \tan ^2(e+f x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \tan ^2(e+f x)+a}{a-b}\right )}{(p+1) (a-b)}}{2 f}\)

Input:

Int[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]
 

Output:

(-(((a + b)*(a + b*Tan[e + f*x]^2)^(1 + p))/(b^2*(1 + p))) - (Hypergeometr 
ic2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x] 
^2)^(1 + p))/((a - b)*(1 + p)) + (a + b*Tan[e + f*x]^2)^(2 + p)/(b^2*(2 + 
p)))/(2*f)
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

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 4153

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], 
 x]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, 
n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio 
nalQ[n]))
 

Maple [F]

\[\int \tan \left (f x +e \right )^{5} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]

Input:

int(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x)
 

Output:

int(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x)
 

Fricas [F]

\[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{5} \,d x } \] Input:

integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")
 

Output:

integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^5, x)
 

Sympy [F]

\[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p} \tan ^{5}{\left (e + f x \right )}\, dx \] Input:

integrate(tan(f*x+e)**5*(a+b*tan(f*x+e)**2)**p,x)
 

Output:

Integral((a + b*tan(e + f*x)**2)**p*tan(e + f*x)**5, x)
 

Maxima [F]

\[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{5} \,d x } \] Input:

integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")
 

Output:

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^5, x)
 

Giac [F]

\[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \tan \left (f x + e\right )^{5} \,d x } \] Input:

integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")
 

Output:

integrate((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^5\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:

int(tan(e + f*x)^5*(a + b*tan(e + f*x)^2)^p,x)
 

Output:

int(tan(e + f*x)^5*(a + b*tan(e + f*x)^2)^p, x)
 

Reduce [F]

\[ \int \tan ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{4} b^{2} p^{2}+\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{4} b^{2} p +\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{2} a b \,p^{2}-\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{2} b^{2} p^{2}-2 \left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{2} b^{2} p -\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} a^{2} p +\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} a b p +2 \left (\tan \left (f x +e \right )^{2} b +a \right )^{p} a b -2 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) a \,b^{2} f \,p^{3}-6 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) a \,b^{2} f \,p^{2}-4 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) a \,b^{2} f p +2 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) b^{3} f \,p^{3}+6 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) b^{3} f \,p^{2}+4 \left (\int \frac {\left (\tan \left (f x +e \right )^{2} b +a \right )^{p} \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b +a}d x \right ) b^{3} f p}{2 b^{2} f p \left (p^{2}+3 p +2\right )} \] Input:

int(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x)
 

Output:

((tan(e + f*x)**2*b + a)**p*tan(e + f*x)**4*b**2*p**2 + (tan(e + f*x)**2*b 
 + a)**p*tan(e + f*x)**4*b**2*p + (tan(e + f*x)**2*b + a)**p*tan(e + f*x)* 
*2*a*b*p**2 - (tan(e + f*x)**2*b + a)**p*tan(e + f*x)**2*b**2*p**2 - 2*(ta 
n(e + f*x)**2*b + a)**p*tan(e + f*x)**2*b**2*p - (tan(e + f*x)**2*b + a)** 
p*a**2*p + (tan(e + f*x)**2*b + a)**p*a*b*p + 2*(tan(e + f*x)**2*b + a)**p 
*a*b - 2*int(((tan(e + f*x)**2*b + a)**p*tan(e + f*x)**3)/(tan(e + f*x)**2 
*b + a),x)*a*b**2*f*p**3 - 6*int(((tan(e + f*x)**2*b + a)**p*tan(e + f*x)* 
*3)/(tan(e + f*x)**2*b + a),x)*a*b**2*f*p**2 - 4*int(((tan(e + f*x)**2*b + 
 a)**p*tan(e + f*x)**3)/(tan(e + f*x)**2*b + a),x)*a*b**2*f*p + 2*int(((ta 
n(e + f*x)**2*b + a)**p*tan(e + f*x)**3)/(tan(e + f*x)**2*b + a),x)*b**3*f 
*p**3 + 6*int(((tan(e + f*x)**2*b + a)**p*tan(e + f*x)**3)/(tan(e + f*x)** 
2*b + a),x)*b**3*f*p**2 + 4*int(((tan(e + f*x)**2*b + a)**p*tan(e + f*x)** 
3)/(tan(e + f*x)**2*b + a),x)*b**3*f*p)/(2*b**2*f*p*(p**2 + 3*p + 2))