3.356 \(\int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx\)
Optimal. Leaf size=40 \[ -\frac{(b \sec (e+f x))^m \, _2F_1\left (3,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
[Out]
-((Hypergeometric2F1[3, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m))
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Rubi [A] time = 0.0439608, antiderivative size = 40, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{2606, 364} \[ -\frac{(b \sec (e+f x))^m \, _2F_1\left (3,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]
[Out]
-((Hypergeometric2F1[3, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m))
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps
\begin{align*} \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(b x)^{-1+m}}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\, _2F_1\left (3,\frac{m}{2};\frac{2+m}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m}\\ \end{align*}
Mathematica [C] time = 21.7939, size = 4177, normalized size = 104.42 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]
[Out]
-(Cot[e + f*x]^5*(b*Sec[e + f*x])^m*((1 - Tan[(e + f*x)/2]^2)^(-1))^(5 + m)*(-1 + Tan[(e + f*x)/2]^2)^5*(-64*A
ppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^2)^m
+ 12*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^
2)^m - AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4*(1 - Tan[(e + f*x)/2]
^2)^m - (2^(5 + m)*AppellF1[1 - m, -m, 1, 2 - m, (1 - Tan[(e + f*x)/2]^2)/2, 1 - Tan[(e + f*x)/2]^2]*(-1 + Tan
[(e + f*x)/2]^2))/(-1 + m) + (12*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/
2]^2*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m - (AppellF1[2, m, -m, 3
, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]
^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m))/(64*f*((-5*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*((1 - Tan[(e + f*x)/2]^2)^(
-1))^(5 + m)*(-1 + Tan[(e + f*x)/2]^2)^4*(-64*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2
]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^2)^m + 12*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/
2]^2]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^2)^m - AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)
/2]^2]*Tan[(e + f*x)/2]^4*(1 - Tan[(e + f*x)/2]^2)^m - (2^(5 + m)*AppellF1[1 - m, -m, 1, 2 - m, (1 - Tan[(e +
f*x)/2]^2)/2, 1 - Tan[(e + f*x)/2]^2]*(-1 + Tan[(e + f*x)/2]^2))/(-1 + m) + (12*AppellF1[1, m, -m, 2, Cot[(e +
f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]^2)^m)/(1
+ Cot[(e + f*x)/2]^2)^m - (AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(
1 - Cot[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m))/64 - ((5 + m)*Sec[(e + f*x)
/2]^2*Tan[(e + f*x)/2]*((1 - Tan[(e + f*x)/2]^2)^(-1))^(6 + m)*(-1 + Tan[(e + f*x)/2]^2)^5*(-64*AppellF1[1, m,
1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^2)^m + 12*AppellF
1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2*(1 - Tan[(e + f*x)/2]^2)^m - Appell
F1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4*(1 - Tan[(e + f*x)/2]^2)^m - (2^(5
+ m)*AppellF1[1 - m, -m, 1, 2 - m, (1 - Tan[(e + f*x)/2]^2)/2, 1 - Tan[(e + f*x)/2]^2]*(-1 + Tan[(e + f*x)/2]
^2))/(-1 + m) + (12*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(1 - Cot
[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m - (AppellF1[2, m, -m, 3, Cot[(e + f*
x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + C
ot[(e + f*x)/2]^2)^m))/64 - (((1 - Tan[(e + f*x)/2]^2)^(-1))^(5 + m)*(-1 + Tan[(e + f*x)/2]^2)^5*(-((2^(5 + m)
*AppellF1[1 - m, -m, 1, 2 - m, (1 - Tan[(e + f*x)/2]^2)/2, 1 - Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e +
f*x)/2])/(-1 + m)) + 64*m*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^
2*Tan[(e + f*x)/2]^3*(1 - Tan[(e + f*x)/2]^2)^(-1 + m) - 12*m*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(
e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^3*(1 - Tan[(e + f*x)/2]^2)^(-1 + m) + m*AppellF1[2, m, -m,
3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^5*(1 - Tan[(e + f*x)/2]^2)^(-1
+ m) - 64*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/
2]*(1 - Tan[(e + f*x)/2]^2)^m + 12*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x
)/2]^2*Tan[(e + f*x)/2]*(1 - Tan[(e + f*x)/2]^2)^m - 2*AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x
)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^3*(1 - Tan[(e + f*x)/2]^2)^m - 64*Tan[(e + f*x)/2]^2*(-((1 - m)*Ap
pellF1[2, m, 2 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + (m*Ap
pellF1[2, 1 + m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2)*(1
- Tan[(e + f*x)/2]^2)^m + 12*Tan[(e + f*x)/2]^2*((m*AppellF1[2, m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*
x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + (m*AppellF1[2, 1 + m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f
*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2)*(1 - Tan[(e + f*x)/2]^2)^m - Tan[(e + f*x)/2]^4*((2*m*Appell
F1[3, m, 1 - m, 4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (2*m*Appe
llF1[3, 1 + m, -m, 4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(1 - Ta
n[(e + f*x)/2]^2)^m - (2^(5 + m)*(((1 - m)*m*AppellF1[2 - m, 1 - m, 1, 3 - m, (1 - Tan[(e + f*x)/2]^2)/2, 1 -
Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2*(2 - m)) - ((1 - m)*AppellF1[2 - m, -m, 2, 3 - m,
(1 - Tan[(e + f*x)/2]^2)/2, 1 - Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 - m))*(-1 + Tan[(e
+ f*x)/2]^2))/(-1 + m) - (m*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*(
1 - Cot[(e + f*x)/2]^2)^m*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]^2)^(-1 + m))/(1 + Cot[(e + f*x)/2]^2)^m + (
12*m*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*(1 - Cot[(e + f*x)/2]^2)^m*Csc[(e + f*x)/2
]*Sec[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]^2)^(-1 + m))/(1 + Cot[(e + f*x)/2]^2)^m + 12*m*AppellF1[1, m, -m, 2,
Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^3*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Cot[(e + f*x)/2]^2
)^(-1 - m)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]^2)^m - m*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e
+ f*x)/2]^2]*Cot[(e + f*x)/2]^5*(1 - Cot[(e + f*x)/2]^2)^m*(1 + Cot[(e + f*x)/2]^2)^(-1 - m)*Csc[(e + f*x)/2]
^2*(1 + Tan[(e + f*x)/2]^2)^m + (12*m*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e +
f*x)/2]^3*(1 - Cot[(e + f*x)/2]^2)^(-1 + m)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/
2]^2)^m - (m*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^5*(1 - Cot[(e + f
*x)/2]^2)^(-1 + m)*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m - (12*AppellF1[1,
m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*(1 - Cot[(e + f*x)/2]^2)^m*Csc[(e + f*x)/
2]^2*(1 + Tan[(e + f*x)/2]^2)^m)/(1 + Cot[(e + f*x)/2]^2)^m + (2*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Co
t[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^3*(1 - Cot[(e + f*x)/2]^2)^m*Csc[(e + f*x)/2]^2*(1 + Tan[(e + f*x)/2]^2)^m)
/(1 + Cot[(e + f*x)/2]^2)^m + (12*Cot[(e + f*x)/2]^2*(1 - Cot[(e + f*x)/2]^2)^m*(-(m*AppellF1[2, m, 1 - m, 3,
Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/2 - (m*AppellF1[2, 1 + m, -m, 3,
Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/2)*(1 + Tan[(e + f*x)/2]^2)^m)/
(1 + Cot[(e + f*x)/2]^2)^m - (Cot[(e + f*x)/2]^4*(1 - Cot[(e + f*x)/2]^2)^m*((-2*m*AppellF1[3, m, 1 - m, 4, Co
t[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/3 - (2*m*AppellF1[3, 1 + m, -m, 4,
Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/3)*(1 + Tan[(e + f*x)/2]^2)^m)/
(1 + Cot[(e + f*x)/2]^2)^m))/64))
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{5} \left ( b\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)^5*(b*sec(f*x+e))^m,x)
[Out]
int(cot(f*x+e)^5*(b*sec(f*x+e))^m,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="maxima")
[Out]
integrate((b*sec(f*x + e))^m*cot(f*x + e)^5, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="fricas")
[Out]
integral((b*sec(f*x + e))^m*cot(f*x + e)^5, x)
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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(f*x+e)**5*(b*sec(f*x+e))**m,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e))^m*cot(f*x + e)^5, x)