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

Optimal. Leaf size=102 \[ -\frac{2^{n-1} \cot (c+d x) \left (\frac{1}{\sec (c+d x)+1}\right )^{n-1} (a \sec (c+d x)+a)^n F_1\left (-\frac{1}{2};n-2,1;\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]

[Out]

-((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)

________________________________________________________________________________________

Rubi [A]  time = 0.0553295, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3889} \[ -\frac{2^{n-1} \cot (c+d x) \left (\frac{1}{\sec (c+d x)+1}\right )^{n-1} (a \sec (c+d x)+a)^n F_1\left (-\frac{1}{2};n-2,1;\frac{1}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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)

Rule 3889

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*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*AppellF1[(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])])/(d*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \cot ^2(c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac{2^{-1+n} F_1\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}\\ \end{align*}

Mathematica [B]  time = 15.3672, size = 2492, normalized size = 24.43 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(-3 + n)*Cos[c + d*x]^2*Csc[(c + d*x)/2]^3*Sec[(c + d*x)/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(a*(1 + Sec
[c + d*x]))^n*((12*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)/(-3*A
ppellF1[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) + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(-Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2] + Hyper
geometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)))/(d*(-(2^(-2 + n)*Csc[(c + d*x)/2]^2*(Cos[(
c + d*x)/2]^2*Sec[c + d*x])^n*((12*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)/(-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) + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(-Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*
x)/2]^2] + Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2))) + 2^(-1 + n)*Cot[(c + d*x)
/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*((12*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]
*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(-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) + (12*Sin[(c + d*x)/2]^2*(-(AppellF1[3/2, n, 2, 5/2, Tan[(
c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3 + (n*AppellF1[3/2, 1 + n, 1, 5/2, T
an[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3))/(-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) + n*(Cos[c +
d*x]*Sec[(c + d*x)/2]^2)^(-1 + n)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c
+ d*x)/2])*(-Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2] + Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*
x)/2]^2]*Tan[(c + d*x)/2]^2) - (12*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*(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])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 3*(-(AppellF1[3/2, n, 2, 5/
2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3 + (n*AppellF1[3/2, 1 + n, 1
, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3) + 2*Tan[(c + d*x)/2]^2
*((-6*AppellF1[5/2, n, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5
+ (3*n*AppellF1[5/2, 1 + n, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)
/2])/5 - n*((-3*AppellF1[5/2, 1 + n, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(
c + d*x)/2])/5 + (3*(1 + n)*AppellF1[5/2, 2 + n, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x
)/2]^2*Tan[(c + d*x)/2])/5))))/(-3*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(Appe
llF1[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)^2 + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Hypergeometric2F1[1/2,
n, 3/2, Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - (Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Hyperge
ometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2] - (1 - Tan[(c + d*x)/2]^2)^(-n)))/2 + (Sec[(c + d*x)/2]^2*Tan[(c
+ d*x)/2]*(-Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2] + (1 - Tan[(c + d*x)/2]^2)^(-n)))/2)) + 2^(-1 +
n)*n*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n)*((12*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)/(-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) + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(-Hyperge
ometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2] + Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*Tan[(c + d*x
)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x])))

________________________________________________________________________________________

Maple [F]  time = 0.252, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{2} \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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