3.355 \(\int \cot ^3(e+f x) (b \sec (e+f x))^m \, dx\)
Optimal. Leaf size=39 \[ \frac{(b \sec (e+f x))^m \, _2F_1\left (2,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
[Out]
(Hypergeometric2F1[2, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m)
________________________________________________________________________________________
Rubi [A] time = 0.0444063, antiderivative size = 39, 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} \[ \frac{(b \sec (e+f x))^m \, _2F_1\left (2,\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
[In]
Int[Cot[e + f*x]^3*(b*Sec[e + f*x])^m,x]
[Out]
(Hypergeometric2F1[2, 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 ^3(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 )^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (2,\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 = 17.4608, size = 3020, normalized size = 77.44 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[e + f*x]^3*(b*Sec[e + f*x])^m,x]
[Out]
-(Cot[e + f*x]^3*(2^(2 + m)*AppellF1[1 - m, -m, 1, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Se
c[(e + f*x)/2]^2]*Cos[e + f*x]*(Csc[(e + f*x)/2]^2)^(1 + m) + (-1 + m)*(AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]
^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + (-
8*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2
, -Tan[(e + f*x)/2]^2])*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m))*(b*Sec[e + f*x])^m*(Cos[(
e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]^2)/(8*f*(-1 + m)*(Csc[(e + f*x)/2]^2)^m*(-(m*(2^(2 + m)*AppellF
1[1 - m, -m, 1, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Cos[e + f*x]*(Csc
[(e + f*x)/2]^2)^(1 + m) + (-1 + m)*(AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f
*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + (-8*AppellF1[1, m, 1 - m, 2, Tan[(e +
f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(Csc[(e + f
*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m))*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2])/(8*(-1
+ m)*(Csc[(e + f*x)/2]^2)^m) - (Sec[(e + f*x)/2]^2*(2^(2 + m)*AppellF1[1 - m, -m, 1, 2 - m, (Cos[e + f*x]*Sec
[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Cos[e + f*x]*(Csc[(e + f*x)/2]^2)^(1 + m) + (-1 + m)*(App
ellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2
]^2))^m*(Sec[(e + f*x)/2]^2)^m + (-8*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + Appel
lF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/
2]^2)^m))*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2])/(8*(-1 + m)*(Csc[(e + f*x)/2]^2)^m) - ((Cos[(e
+ f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]^2*(-(2^(2 + m)*(1 + m)*AppellF1[1 - m, -m, 1, 2 - m, (Cos[e + f*
x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Cos[e + f*x]*Cot[(e + f*x)/2]*(Csc[(e + f*x)/2]^2)^
(1 + m)) - 2^(2 + m)*AppellF1[1 - m, -m, 1, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e +
f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*Sin[e + f*x] + 2^(2 + m)*Cos[e + f*x]*(Csc[(e + f*x)/2]^2)^(1 + m)*(-(
((1 - m)*m*AppellF1[2 - m, 1 - m, 1, 3 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]
^2]*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x])/2 + (Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2))/(2 - m)) +
((1 - m)*AppellF1[2 - m, -m, 2, 3 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]^2]*(
-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]))/(2 - m)) + (-1 + m)*(m
*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^3*(-(Cos[e + f*x]*Csc[(e + f*
x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m - 2*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e +
f*x)/2]^3*Csc[(e + f*x)/2]^2*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + Cot[(e + f*x)/2]
^4*(-(Cos[e + f*x]*Csc[(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)*(Sec[(e + f*x)/2]^2)^m - m*(-8*AppellF1[1, m, 1 - m, 2, Tan[(e + f
*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cot[(e + f*x)
/2]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + m*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -
Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^(-1 + m)*(Sec[(e + f*x)/2]^2)^m*(C
os[e + f*x]*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2 + Csc[(e + f*x)/2]^2*Sin[e + f*x]) + m*(-8*AppellF1[1, m, 1 -
m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2
])*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + m)*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos
[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m*((
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 -
8*(-((1 - m)*AppellF1[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*AppellF1[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)))))/(8*(-1 + m)*(Csc[(e + f*x)/2]^2)^m) - (m*(2^(2 + m)*AppellF1[1 - m, -m, 1, 2 - m, (Cos[e + f*
x]*Sec[(e + f*x)/2]^2)/2, Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Cos[e + f*x]*(Csc[(e + f*x)/2]^2)^(1 + m) + (-1 + m
)*(AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^4*(-(Cos[e + f*x]*Csc[(e +
f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + (-8*AppellF1[1, m, 1 - m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] +
AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e +
f*x)/2]^2)^m))*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m)*Tan[(e + f*x)/2]^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]
*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(8*(-1 + m)*(Csc[(e + f*x)/2]^2)^m)))
________________________________________________________________________________________
Maple [F] time = 0.386, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{3} \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)^3*(b*sec(f*x+e))^m,x)
[Out]
int(cot(f*x+e)^3*(b*sec(f*x+e))^m,x)
________________________________________________________________________________________
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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*sec(f*x+e))^m,x, algorithm="maxima")
[Out]
integrate((b*sec(f*x + e))^m*cot(f*x + e)^3, x)
________________________________________________________________________________________
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 )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*sec(f*x+e))^m,x, algorithm="fricas")
[Out]
integral((b*sec(f*x + e))^m*cot(f*x + e)^3, x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)**3*(b*sec(f*x+e))**m,x)
[Out]
Integral((b*sec(e + f*x))**m*cot(e + f*x)**3, x)
________________________________________________________________________________________
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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(f*x+e)^3*(b*sec(f*x+e))^m,x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e))^m*cot(f*x + e)^3, x)