3.4.56 \(\int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx\) [356]
Optimal. Leaf size=40 \[ -\frac {\, _2F_1\left (3,\frac {m}{2};\frac {2+m}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]
[Out]
-hypergeom([3, 1/2*m],[1+1/2*m],sec(f*x+e)^2)*(b*sec(f*x+e))^m/f/m
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {2686, 371}
\begin {gather*} -\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} \end {gather*}
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 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 2686
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])
Rubi steps
\begin {align*} \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx &=\frac {b \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 22.17, size = 2138, normalized size = 53.45 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]
[Out]
(Cos[e + f*x]*Sec[(e + f*x)/2]^2*((1 + Cos[e + f*x])*Hypergeometric2F1[1, 1 - m, 2 - m, Cos[e + f*x]] - (2^m*H
ypergeometric2F1[1 - m, 1 - m, 2 - m, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2])/(Sec[(e + f*x)/2]^2)^m)*(b*Sec[e +
f*x])^m)/(4*f*(-1 + m)) + (3*Cot[(e + f*x)/2]*Cot[e + f*x]*Csc[e + f*x]^4*(4*AppellF1[1, m, -m, 2, Cot[(e + f
*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^6*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(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]^8*(-(Cos[e + f*x]*Csc[(e +
f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^m + (AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 4*Ap
pellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cot[(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)/(2*f*(-6*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e
+ f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*m*Appe
llF1[2, m, 1 - m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x
)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*AppellF1[2, m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot
[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - 3*m*AppellF1[2, 1 + m,
-m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^10*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Se
c[(e + f*x)/2]^2)^(1 + m) - m*AppellF1[3, m, 1 - m, 4, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/
2]^12*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) - m*AppellF1[3, 1 + m, -m, 4, Cot[(e
+ f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^12*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2
]^2)^(1 + m) + 3*m*AppellF1[2, m, 1 - m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 +
m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 3*AppellF1[2, m, -m, 3, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cs
c[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 3*m*AppellF1[2, 1 + m, -m, 3, Tan[(e + f*x)/2]
^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + 6*AppellF1[1, m, -
m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(Csc[(e + f*x)/2]^2)^(1 + m)*(Cos[e + f*x]*S
ec[(e + f*x)/2]^2)^m + m*AppellF1[3, m, 1 - m, 4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2
)^m*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m + m*AppellF1[3, 1 + m, -m, 4, Tan[(e + f*x)/2]^2, -
Tan[(e + f*x)/2]^2]*(Csc[(e + f*x)/2]^2)^m*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^m)) + (4*Cot[(
e + f*x)/2]*Cot[e + f*x]*Csc[e + f*x]^2*(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 + 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)/
(f*(2*AppellF1[1, m, -m, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^6*(-(Cos[e + f*x]*Csc[(e
+ f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + 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]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^m*(Sec[(e + f*x)/2]^2)^(1 + m) + m*AppellF1[2, 1
+ m, -m, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^8*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^
m*(Sec[(e + f*x)/2]^2)^(1 + m) - 2*AppellF1[1, m, -m, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Csc[(e + f*
x)/2]^2)^(1 + 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]*(Csc[(e + f*x)/2]^2)^m*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1 + m)*Sec[e + f*x] - m*AppellF1[2, 1 +
m, -m, 3, 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]))
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\cot ^{5}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx\]
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{5}{\left (e + f x \right )}\, dx \end {gather*}
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]
Integral((b*sec(e + f*x))**m*cot(e + f*x)**5, x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e + f*x)^5*(b/cos(e + f*x))^m,x)
[Out]
int(cot(e + f*x)^5*(b/cos(e + f*x))^m, x)
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