3.1.68 \(\int ((b \csc (c+d x))^p)^n \, dx\) [68]

Optimal. Leaf size=80 \[ \frac {\cos (c+d x) \left ((b \csc (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\sin ^2(c+d x)\right ) \sin (c+d x)}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \]

[Out]

cos(d*x+c)*((b*csc(d*x+c))^p)^n*hypergeom([1/2, -1/2*n*p+1/2],[-1/2*n*p+3/2],sin(d*x+c)^2)*sin(d*x+c)/d/(-n*p+
1)/(cos(d*x+c)^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4208, 3857, 2722} \begin {gather*} \frac {\sin (c+d x) \cos (c+d x) \left ((b \csc (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\sin ^2(c+d x)\right )}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b*Csc[c + d*x])^p)^n,x]

[Out]

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

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3857

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 71, normalized size = 0.89 \begin {gather*} -\frac {\cos (c+d x) \left ((b \csc (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {3}{2};\cos ^2(c+d x)\right ) \sin (c+d x) \sin ^2(c+d x)^{\frac {1}{2} (-1+n p)}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b*Csc[c + d*x])^p)^n,x]

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (\left (b \csc \left (d x +c \right )\right )^{p}\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*csc(d*x+c))^p)^n,x)

[Out]

int(((b*csc(d*x+c))^p)^n,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(((b*csc(d*x+c))^p)^n,x, algorithm="maxima")

[Out]

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

[Out]

integral(((b*csc(d*x + c))^p)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*csc(d*x+c))**p)**n,x)

[Out]

Integral(((b*csc(c + d*x))**p)**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(((b*csc(d*x+c))^p)^n,x, algorithm="giac")

[Out]

integrate(((b*csc(d*x + c))^p)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left ({\left (\frac {b}{\sin \left (c+d\,x\right )}\right )}^p\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b/sin(c + d*x))^p)^n,x)

[Out]

int(((b/sin(c + d*x))^p)^n, x)

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