3.280 \(\int \sqrt{d \cos (a+b x)} \csc ^p(a+b x) \, dx\)

Optimal. Leaf size=76 \[ \frac{d \sqrt [4]{\cos ^2(a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt{d \cos (a+b x)}} \]

[Out]

(d*(Cos[a + b*x]^2)^(1/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[1/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/
(b*(1 - p)*Sqrt[d*Cos[a + b*x]])

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Rubi [A]  time = 0.091963, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2587, 2577} \[ \frac{d \sqrt [4]{\cos ^2(a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^p,x]

[Out]

(d*(Cos[a + b*x]^2)^(1/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[1/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/
(b*(1 - p)*Sqrt[d*Cos[a + b*x]])

Rule 2587

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[b^2*(b*Cos[e
+ f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1), Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e,
f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2577

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

Rubi steps

\begin{align*} \int \sqrt{d \cos (a+b x)} \csc ^p(a+b x) \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int \sqrt{d \cos (a+b x)} \sin ^{-p}(a+b x) \, dx\\ &=\frac{d \sqrt [4]{\cos ^2(a+b x)} \csc ^{-1+p}(a+b x) \, _2F_1\left (\frac{1}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.171021, size = 70, normalized size = 0.92 \[ -\frac{2 (d \cos (a+b x))^{3/2} \sin ^2(a+b x)^{\frac{p-1}{2}} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{3}{4},\frac{p+1}{2};\frac{7}{4};\cos ^2(a+b x)\right )}{3 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^p,x]

[Out]

(-2*(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[3/4, (1 + p)/2, 7/4, Cos[a + b*x]^2]*(Sin[a
+ b*x]^2)^((-1 + p)/2))/(3*b*d)

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Maple [F]  time = 0.31, size = 0, normalized size = 0. \begin{align*} \int \sqrt{d\cos \left ( bx+a \right ) } \left ( \csc \left ( bx+a \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x)

[Out]

int((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \cos{\left (a + b x \right )}} \csc ^{p}{\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))**(1/2)*csc(b*x+a)**p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*csc(b*x+a)^p,x, algorithm="giac")

[Out]

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