\(\int \cot ^5(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [366]

Optimal result

Integrand size = 23, antiderivative size = 218 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {(2 a+b-b p) \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{4 a^2 f}-\frac {\cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{4 a f}-\frac {\left (2 a^2-2 a b p-b^2 (1-p) p\right ) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{4 a^3 f (1+p)}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right ) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{2 (a-b) f (1+p)} \] Output:

1/4*(-b*p+2*a+b)*cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(p+1)/a^2/f-1/4*cot(f*x+e 
)^4*(a+b*tan(f*x+e)^2)^(p+1)/a/f-1/4*(2*a^2-2*a*b*p-b^2*(1-p)*p)*hypergeom 
([1, p+1],[2+p],(a+b*tan(f*x+e)^2)/a)*(a+b*tan(f*x+e)^2)^(p+1)/a^3/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)
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.79 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\left (b+a \cot ^2(e+f x)\right ) \left (-2 a^3 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (a (1+p) \cot ^2(e+f x) \left (-2 a+b (-1+p)+a \cot ^2(e+f x)\right )+\left (2 a^2-2 a b p+b^2 (-1+p) p\right ) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \tan ^2(e+f x)}{a}\right )\right )\right ) \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p}{4 a^3 (a-b) f (1+p)} \] Input:

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

Output:

-1/4*((b + a*Cot[e + f*x]^2)*(-2*a^3*Hypergeometric2F1[1, 1 + p, 2 + p, (a 
 + b*Tan[e + f*x]^2)/(a - b)] + (a - b)*(a*(1 + p)*Cot[e + f*x]^2*(-2*a + 
b*(-1 + p) + a*Cot[e + f*x]^2) + (2*a^2 - 2*a*b*p + b^2*(-1 + p)*p)*Hyperg 
eometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]))*Tan[e + f*x]^2*(a 
 + b*Tan[e + f*x]^2)^p)/(a^3*(a - b)*f*(1 + p))
 

Rubi [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4153, 354, 114, 168, 174, 75, 78}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\cot ^3(e+f x) \left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}d\tan ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

(-1/2*(Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(1 + p))/a - (-(((2*a + b - b 
*p)*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(1 + p))/a) - ((2*a^2*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)) - ((2*a^2 - 2*a*b*p - b^2*(1 - p)*p)*Hyperg 
eometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x] 
^2)^(1 + p))/(a*(1 + p)))/a)/(2*a))/(2*f)
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(cot(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*cot(f*x + e)^5, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(cot(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*cot(f*x + e)^5, x)
                                                                                    
                                                                                    
 

Giac [F]

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

integrate(cot(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*cot(f*x + e)^5, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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