3.1.28 \(\int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx\) [28]

Optimal. Leaf size=48 \[ -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a+a \csc (c+d x)}} \]

[Out]

-2*a*cot(d*x+c)*hypergeom([1/2, 1-n],[3/2],1-csc(d*x+c))/d/(a+a*csc(d*x+c))^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3891, 67} \begin {gather*} -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a \csc (c+d x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^n*Sqrt[a + a*Csc[c + d*x]],x]

[Out]

(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a + a*Csc[c + d*x]])

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 3891

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[a^2*d*(
Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x]
, x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.68, size = 48, normalized size = 1.00 \begin {gather*} -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a (1+\csc (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^n*Sqrt[a + a*Csc[c + d*x]],x]

[Out]

(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a*(1 + Csc[c + d*x])])

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (\csc ^{n}\left (d x +c \right )\right ) \sqrt {a +a \csc \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x)

[Out]

int(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),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(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^n, 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(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**n*(a+a*csc(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*(csc(c + d*x) + 1))*csc(c + d*x)**n, 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(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/sin(c + d*x))^(1/2)*(1/sin(c + d*x))^n,x)

[Out]

int((a + a/sin(c + d*x))^(1/2)*(1/sin(c + d*x))^n, x)

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