3.364 \(\int (d \cos (a+b x))^n \csc ^2(a+b x) \, dx\)

Optimal. Leaf size=69 \[ -\frac{\sqrt{\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0397905, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2576} \[ -\frac{\sqrt{\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*Csc[a + b*x]^2,x]

[Out]

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

Rule 2576

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^(2*IntPar
t[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/
2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a,
b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps

\begin{align*} \int (d \cos (a+b x))^n \csc ^2(a+b x) \, dx &=-\frac{(d \cos (a+b x))^{1+n} \csc (a+b x) \, _2F_1\left (\frac{3}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right ) \sqrt{\sin ^2(a+b x)}}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.176787, size = 80, normalized size = 1.16 \[ \frac{d \csc (a+b x) \left (-\cot ^2(a+b x)\right )^{\frac{1-n}{2}} (d \cos (a+b x))^{n-1} \, _2F_1\left (\frac{1-n}{2},1-\frac{n}{2};2-\frac{n}{2};\csc ^2(a+b x)\right )}{b (n-2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^2,x]

[Out]

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

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Maple [F]  time = 0.359, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( \csc \left ( bx+a \right ) \right ) ^{2}\, 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)^2,x)

[Out]

int((d*cos(b*x+a))^n*csc(b*x+a)^2,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 )^{2}\,{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)^2,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a)^2, 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 )^{2}, 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)^2,x, algorithm="fricas")

[Out]

integral((d*cos(b*x + a))^n*csc(b*x + a)^2, 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 ^{2}{\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)**2,x)

[Out]

Integral((d*cos(a + b*x))**n*csc(a + b*x)**2, 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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^n*csc(b*x + a)^2, x)