3.496 \(\int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx\)
Optimal. Leaf size=48 \[ \frac{(b \sec (e+f x))^{n+3} \, _2F_1\left (2,\frac{n+3}{2};\frac{n+5}{2};\sec ^2(e+f x)\right )}{b^3 f (n+3)} \]
[Out]
(Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3 + n))/(b^3*f*(3 + n))
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Rubi [A] time = 0.0516906, antiderivative size = 48, 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 =
{2622, 364} \[ \frac{(b \sec (e+f x))^{n+3} \, _2F_1\left (2,\frac{n+3}{2};\frac{n+5}{2};\sec ^2(e+f x)\right )}{b^3 f (n+3)} \]
Antiderivative was successfully verified.
[In]
Int[Csc[e + f*x]^3*(b*Sec[e + f*x])^n,x]
[Out]
(Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3 + n))/(b^3*f*(3 + n))
Rule 2622
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
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 \csc ^3(e+f x) (b \sec (e+f x))^n \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^{2+n}}{\left (-1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{b^3 f}\\ &=\frac{\, _2F_1\left (2,\frac{3+n}{2};\frac{5+n}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^{3+n}}{b^3 f (3+n)}\\ \end{align*}
Mathematica [C] time = 16.8315, size = 2113, normalized size = 44.02 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Csc[e + f*x]^3*(b*Sec[e + f*x])^n,x]
[Out]
(Csc[e + f*x]^3*(b*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*((2^(1 + n)*AppellF1[1 - n, -n, 1, 2 -
n, (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]*Sec[(e + f*x)/2]^2)/(-1
+ n) - (AppellF1[1, n, -n, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(-(Cos[e + f*x]*Csc[
(e + f*x)/2]^2))^n*(Sec[(e + f*x)/2]^2)^n)/(Csc[(e + f*x)/2]^2)^n + AppellF1[1, n, -n, 2, Tan[(e + f*x)/2]^2,
-Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2))/(8*f*(((Cos[(e + f*x)/2]^2*Sec[e
+ f*x])^n*(AppellF1[1, n, -n, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*(Csc[(e + f*x)/2]^
2)^(1 - n)*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^n*(Sec[(e + f*x)/2]^2)^n - (n*AppellF1[1, n, -n, 2, Cot[(e + f
*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]*(-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^n*(Sec[(e + f*x)/2]^2)^n)
/(Csc[(e + f*x)/2]^2)^n - (n*AppellF1[1, n, -n, 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))^n*(Sec[(e + f*x)/2]^2)^n)/(Csc[(e + f*x)/2]^2)^n - (Cot[(e + f*x)/2]^2*(
-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^n*(-(n*AppellF1[2, n, 1 - n, 3, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*C
ot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/2 - (n*AppellF1[2, 1 + n, -n, 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)^n)/(Csc[(e + f*x)/2]^2)^n - (2^(1 + n)*AppellF1[1
- n, -n, 1, 2 - n, (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*S
in[e + f*x])/(-1 + n) - (n*AppellF1[1, n, -n, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(
-(Cos[e + f*x]*Csc[(e + f*x)/2]^2))^(-1 + n)*(Sec[(e + f*x)/2]^2)^n*(Cos[e + f*x]*Cot[(e + f*x)/2]*Csc[(e + f*
x)/2]^2 + Csc[(e + f*x)/2]^2*Sin[e + f*x]))/(Csc[(e + f*x)/2]^2)^n + (2^(1 + n)*AppellF1[1 - n, -n, 1, 2 - n,
(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]*Sec[(e + f*x)/2]^2*Tan[(e +
f*x)/2])/(-1 + n) + AppellF1[1, n, -n, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e
+ f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2] + (Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*((n*App
ellF1[2, n, 1 - n, 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 + (n*App
ellF1[2, 1 + n, -n, 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) + n*Ap
pellF1[1, n, -n, 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*Tan[(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^(1 +
n)*Cos[e + f*x]*Sec[(e + f*x)/2]^2*(-(((1 - n)*n*AppellF1[2 - n, 1 - n, 1, 3 - n, (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 - n)) + ((1 - n)*AppellF1[2 - n, -n, 2, 3 - n, (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*Ta
n[(e + f*x)/2]))/(2 - n)))/(-1 + n)))/8 + (n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*((2^(1 + n)*AppellF1[1
- n, -n, 1, 2 - n, (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]*Sec[(e
+ f*x)/2]^2)/(-1 + n) - (AppellF1[1, n, -n, 2, Cot[(e + f*x)/2]^2, -Cot[(e + f*x)/2]^2]*Cot[(e + f*x)/2]^2*(-(
Cos[e + f*x]*Csc[(e + f*x)/2]^2))^n*(Sec[(e + f*x)/2]^2)^n)/(Csc[(e + f*x)/2]^2)^n + AppellF1[1, n, -n, 2, Tan
[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*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))
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Maple [F] time = 0.397, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{3} \left ( b\sec \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(f*x+e)^3*(b*sec(f*x+e))^n,x)
[Out]
int(csc(f*x+e)^3*(b*sec(f*x+e))^n,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 )^{n} \csc \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="maxima")
[Out]
integrate((b*sec(f*x + e))^n*csc(f*x + e)^3, 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 )^{n} \csc \left (f x + e\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="fricas")
[Out]
integral((b*sec(f*x + e))^n*csc(f*x + e)^3, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)**3*(b*sec(f*x+e))**n,x)
[Out]
Integral((b*sec(e + f*x))**n*csc(e + f*x)**3, x)
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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 )^{n} \csc \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="giac")
[Out]
integrate((b*sec(f*x + e))^n*csc(f*x + e)^3, x)