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]

hypergeom([2, 1/2*m],[1+1/2*m],sec(f*x+e)^2)*(b*sec(f*x+e))^m/f/m

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 39, 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 = {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 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])

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])

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 = 12.23, size = 815, normalized size = 20.90 \[ \frac {\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (2^m \, _2F_1\left (1-m,1-m;2-m;\frac {1}{2} \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m}-(\cos (e+f x)+1) \, _2F_1(1,1-m;2-m;\cos (e+f x))\right ) (b \sec (e+f x))^m}{4 f (m-1)}-\frac {2 \cot \left (\frac {1}{2} (e+f x)\right ) \cot (e+f x) \csc ^2(e+f x) \left (F_1\left (1;m,-m;2;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^4\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^m+F_1\left (1;m,-m;2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^m\right ) (b \sec (e+f x))^m}{f \left (-m F_1\left (2;m,1-m;3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{m+1} \sec (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m-m F_1\left (2;m+1,-m;3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{m+1} \sec (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )^m-2 F_1\left (1;m,-m;2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \csc ^2\left (\frac {1}{2} (e+f x)\right )^{m+1}+m F_1\left (2;m,1-m;3;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^8\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{m+1}+m F_1\left (2;m+1,-m;3;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^8\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{m+1}+2 F_1\left (1;m,-m;2;\cot ^2\left (\frac {1}{2} (e+f x)\right ),-\cot ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cot ^6\left (\frac {1}{2} (e+f x)\right ) \left (-\cos (e+f x) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{m+1}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^3*(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*Hypergeometric2F1[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)) - (2*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]))

________________________________________________________________________________________

fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{3}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^3*(b/cos(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^3*(b/cos(e + f*x))^m, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \]

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)

________________________________________________________________________________________