3.4.58 \(\int (d \cos (a+b x))^n \csc (a+b x) \, dx\) [358]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {2645, 371} \begin {gather*} -\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)} \end {gather*}

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 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 2645

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])

Rubi steps

\begin {align*} \int (d \cos (a+b x))^n \csc (a+b x) \, dx &=-\frac {\text {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*}

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Mathematica [A]
time = 0.09, size = 52, normalized size = 1.06 \begin {gather*} -\frac {\cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left (1,\frac {1+n}{2};1+\frac {1+n}{2};\cos ^2(a+b x)\right )}{b (1+n)} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (b x +a \right )\right )^{n} \csc \left (b x +a \right )\, dx\]

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 = 0.00, 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((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 = 0.00, 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \cos {\left (a + b x \right )}\right )^{n} \csc {\left (a + b x \right )}\, dx \end {gather*}

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 = 0.00, 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((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{\sin \left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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