3.358 \(\int (d \cos (a+b x))^n \csc (a+b x) \, dx\)

Optimal. Leaf size=49 \[ -\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

[Out]

-(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2])/(b*d*(1 + n)))

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Rubi [A]  time = 0.0404455, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2565, 364} \[ -\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*Csc[a + b*x],x]

[Out]

-(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2])/(b*d*(1 + n)))

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 (d \cos (a+b x))^n \csc (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^n}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac{(d \cos (a+b x))^{1+n} \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right )}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0330812, size = 52, normalized size = 1.06 \[ -\frac{\cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;\cos ^2(a+b x)\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x],x]

[Out]

-((Cos[a + b*x]*(d*Cos[a + b*x])^n*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Cos[a + b*x]^2])/(b*(1 + n))
)

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Maple [F]  time = 0.385, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n}\csc \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^n*csc(b*x+a),x)

[Out]

int((d*cos(b*x+a))^n*csc(b*x+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a),x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a),x, algorithm="fricas")

[Out]

integral((d*cos(b*x + a))^n*csc(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos{\left (a + b x \right )}\right )^{n} \csc{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**n*csc(b*x+a),x)

[Out]

Integral((d*cos(a + b*x))**n*csc(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*csc(b*x+a),x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a), x)