\(\int \cot ^4(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [372]

Optimal result

Integrand size = 23, antiderivative size = 84 \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},1,-p,-\frac {1}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {a+b \tan ^2(e+f x)}{a}\right )^{-p}}{3 f} \] Output:

-1/3*AppellF1(-3/2,1,-p,-1/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*cot(f*x+e)^3 
*(a+b*tan(f*x+e)^2)^p/f/(((a+b*tan(f*x+e)^2)/a)^p)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1887\) vs. \(2(84)=168\).

Time = 6.42 (sec) , antiderivative size = 1887, normalized size of antiderivative = 22.46 \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx =\text {Too large to display} \] Input:

Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]
 

Output:

(2*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]* 
(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p) + (Cot[e + f*x] 
^3*(a + b*Tan[e + f*x]^2)^p*(-a - b*Tan[e + f*x]^2 - ((3*a + b*(-1 + 2*p)) 
*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2) 
/(1 + (b*Tan[e + f*x]^2)/a)^p))/(3*a*f) + (3*a*AppellF1[1/2, -p, 1, 3/2, - 
((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a + b* 
Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x] 
^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e 
+ f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + 
f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, -p, 
1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(a + b*Ta 
n[e + f*x]^2)^(-1 + p))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2 
)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + 
f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f* 
x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*AppellF1[1/2, -p, 1, 3/ 
2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(a + b*Tan[e + 
 f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e 
 + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), 
-Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -T 
an[e + f*x]^2])*Tan[e + f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*T...
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4153, 395, 394}

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.

\(\displaystyle \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

\(\displaystyle -\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (-\frac {3}{2},1,-p,-\frac {1}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{3 f}\)

Input:

Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]
 

Output:

-1/3*(AppellF1[-3/2, 1, -p, -1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a) 
]*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p 
)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

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]

\[\int \cot \left (f x +e \right )^{4} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]

Input:

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)
 

Output:

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)
 

Fricas [F]

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

integrate(cot(f*x+e)^4*(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)^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)**4*(a+b*tan(f*x+e)**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cot(f*x+e)^4*(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)^4, x)
 

Giac [F]

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

integrate(cot(f*x+e)^4*(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)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:

int(cot(e + f*x)^4*(a + b*tan(e + f*x)^2)^p,x)
 

Output:

int(cot(e + f*x)^4*(a + b*tan(e + f*x)^2)^p, x)
 

Reduce [F]

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

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)
 

Output:

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)