\(\int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx\) [227]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {2^{-1+n} \operatorname {AppellF1}\left (-\frac {1}{2},-2+n,1,\frac {1}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \cot (c+d x) \left (\frac {1}{1+\sec (c+d x)}\right )^{-1+n} (a+a \sec (c+d x))^n}{d} \] Output:

-2^(-1+n)*AppellF1(-1/2,1,-2+n,1/2,(a-a*sec(d*x+c))/(a+a*sec(d*x+c)),-(a-a 
*sec(d*x+c))/(a+a*sec(d*x+c)))*cot(d*x+c)*(1/(1+sec(d*x+c)))^(-1+n)*(a+a*s 
ec(d*x+c))^n/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(893\) vs. \(2(102)=204\).

Time = 2.66 (sec) , antiderivative size = 893, normalized size of antiderivative = 8.75 \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx =\text {Too large to display} \] Input:

Integrate[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]
 

Output:

((a*(1 + Sec[c + d*x]))^n*(-((2^n*Cot[(c + d*x)/2]*Hypergeometric2F1[-1/2, 
 n, 1/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + 
 d*x)/2]^2*Sec[c + d*x])^n)/(1 + Sec[c + d*x])^n) + (2^n*Hypergeometric2F1 
[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos 
[(c + d*x)/2]^2*Sec[c + d*x])^n*Tan[(c + d*x)/2])/(1 + Sec[c + d*x])^n - ( 
60*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[( 
c + d*x)/2]^2*Cos[c + d*x]*Sin[c + d*x]*(3*AppellF1[1/2, n, 1, 3/2, Tan[(c 
 + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, n, 2, 5/2, Tan[(c + 
d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/2, 1 + n, 1, 5/2, Tan[(c + 
d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(45*AppellF1[1/2, n, 
 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 
+ 2*n - 2*n*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, n, 1, 3/2, 
Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3 
/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + 
n)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*n*AppellF1[3/2, 1 + n, 1, 5/2, Tan 
[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + 
Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, n, 3, 7/2, Tan[(c + d*x)/2]^2, -Ta 
n[(c + d*x)/2]^2] - 2*n*AppellF1[5/2, 1 + n, 2, 7/2, Tan[(c + d*x)/2]^2, - 
Tan[(c + d*x)/2]^2] + n*(1 + n)*AppellF1[5/2, 2 + n, 1, 7/2, Tan[(c + d*x) 
/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]...
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3042, 4377}

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.

\(\displaystyle \int \cot ^2(c+d x) (a \sec (c+d x)+a)^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^n}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx\)

\(\Big \downarrow \) 4377

\(\displaystyle -\frac {2^{n-1} \cot (c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n-1} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (-\frac {1}{2},n-2,1,\frac {1}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d}\)

Input:

Int[Cot[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]
 

Output:

-((2^(-1 + n)*AppellF1[-1/2, -2 + n, 1, 1/2, -((a - a*Sec[c + d*x])/(a + a 
*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d*x]*( 
(1 + Sec[c + d*x])^(-1))^(-1 + n)*(a + a*Sec[c + d*x])^n)/d)
 

Defintions of rubi rules used

rule 3042

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

rule 4377

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_))^(n_), x_Symbol] :> Simp[(-2^(m + n + 1))*(e*Cot[c + d*x])^(m + 1)*((a 
+ b*Csc[c + d*x])^n/(d*e*(m + 1)))*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*App 
ellF1[(m + 1)/2, m + n, 1, (m + 3)/2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + 
d*x]), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])], x] /; FreeQ[{a, b, c, d, 
 e, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \cot \left (d x +c \right )^{2} \left (a +a \sec \left (d x +c \right )\right )^{n}d x\]

Input:

int(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x)
 

Output:

int(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x)
 

Fricas [F]

\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x, algorithm="fricas")
 

Output:

integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)
 

Sympy [F]

\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \cot ^{2}{\left (c + d x \right )}\, dx \] Input:

integrate(cot(d*x+c)**2*(a+a*sec(d*x+c))**n,x)
 

Output:

Integral((a*(sec(c + d*x) + 1))**n*cot(c + d*x)**2, x)
 

Maxima [F]

\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x, algorithm="maxima")
 

Output:

integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)
 

Giac [F]

\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x, algorithm="giac")
 

Output:

integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \] Input:

int(cot(c + d*x)^2*(a + a/cos(c + d*x))^n,x)
 

Output:

int(cot(c + d*x)^2*(a + a/cos(c + d*x))^n, x)
 

Reduce [F]

\[ \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx=\int \left (\sec \left (d x +c \right ) a +a \right )^{n} \cot \left (d x +c \right )^{2}d x \] Input:

int(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x)
 

Output:

int((sec(c + d*x)*a + a)**n*cot(c + d*x)**2,x)