\(\int (d \cot (e+f x))^m (a+b \tan ^2(e+f x))^p \, dx\) [424]

Optimal result

Integrand size = 25, antiderivative size = 108 \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1-m}{2},1,-p,\frac {3-m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {a+b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)} \] Output:

AppellF1(1/2-1/2*m,1,-p,3/2-1/2*m,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*(d*cot( 
f*x+e))^m*tan(f*x+e)*(a+b*tan(f*x+e)^2)^p/f/(1-m)/(((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. \(265\) vs. \(2(108)=216\).

Time = 1.33 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.45 \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {a (-3+m) \operatorname {AppellF1}\left (\frac {1-m}{2},-p,1,\frac {3-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p}{f (-1+m) \left (-2 b p \operatorname {AppellF1}\left (\frac {3-m}{2},1-p,1,\frac {5-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 a \operatorname {AppellF1}\left (\frac {3-m}{2},-p,2,\frac {5-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a (-3+m) \operatorname {AppellF1}\left (\frac {1-m}{2},-p,1,\frac {3-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cot ^2(e+f x)\right )} \] Input:

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

Output:

-((a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a 
), -Tan[e + f*x]^2]*Cos[e + f*x]^2*Cot[e + f*x]*(d*Cot[e + f*x])^m*(a + b* 
Tan[e + f*x]^2)^p)/(f*(-1 + m)*(-2*b*p*AppellF1[(3 - m)/2, 1 - p, 1, (5 - 
m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*a*AppellF1[(3 - m)/2, 
-p, 2, (5 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + a*(-3 + m)*A 
ppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x 
]^2]*Cot[e + f*x]^2)))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4157, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

rule 4157

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + ( 
f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[(d*Cot[e + f*x])^FracPart[m]*(Tan 
[e + f*x]/d)^FracPart[m]   Int[(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d 
)^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*tan(f*x + e)^2 + a)^p*(d*cot(f*x + e))^m, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((b*tan(f*x + e)^2 + a)^p*(d*cot(f*x + e))^m, x)
 

Giac [F]

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

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

Output:

integrate((b*tan(f*x + e)^2 + a)^p*(d*cot(f*x + e))^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d**m*int((tan(e + f*x)**2*b + a)**p*cot(e + f*x)**m,x)