3.228 \(\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx\)

Optimal. Leaf size=106 \[ -\frac {2^{n-3} \cot ^3(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n-3} (a \sec (c+d x)+a)^n F_1\left (-\frac {3}{2};n-4,1;-\frac {1}{2};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d} \]

[Out]

-1/3*2^(-3+n)*AppellF1(-3/2,-4+n,1,-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)^3*(1/(1+sec(d*x+c)))^(-3+n)*(a+a*sec(d*x+c))^n/d

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, 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} \[ -\frac {2^{n-3} \cot ^3(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n-3} (a \sec (c+d x)+a)^n F_1\left (-\frac {3}{2};n-4,1;-\frac {1}{2};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]

[Out]

-(2^(-3 + n)*AppellF1[-3/2, -4 + 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]^3*((1 + Sec[c + d*x])^(-1))^(-3 + n)*(a + a*Sec[c + d*x])^n)/(3*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 ^4(c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac {2^{-3+n} F_1\left (-\frac {3}{2};-4+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 ^3(c+d x) \left (\frac {1}{1+\sec (c+d x)}\right )^{-3+n} (a+a \sec (c+d x))^n}{3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.70, size = 0, normalized size = 0.00 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]

[Out]

Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n, x]

________________________________________________________________________________________

fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)

________________________________________________________________________________________

maple [F]  time = 1.13, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{4}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)

[Out]

int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(a + a/cos(c + d*x))^n,x)

[Out]

int(cot(c + d*x)^4*(a + a/cos(c + d*x))^n, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+a*sec(d*x+c))**n,x)

[Out]

Timed out

________________________________________________________________________________________