3.10 \(\int \csc ^{\frac {5}{2}}(a+b x) \, dx\)

Optimal. Leaf size=67 \[ \frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b} \]

[Out]

-2/3*cos(b*x+a)*csc(b*x+a)^(3/2)/b-2/3*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticF
(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*csc(b*x+a)^(1/2)*sin(b*x+a)^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2641} \[ \frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b}-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^(5/2),x]

[Out]

(-2*Cos[a + b*x]*Csc[a + b*x]^(3/2))/(3*b) + (2*Sqrt[Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a
+ b*x]])/(3*b)

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int \csc ^{\frac {5}{2}}(a+b x) \, dx &=-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {1}{3} \int \sqrt {\csc (a+b x)} \, dx\\ &=-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {1}{3} \left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx\\ &=-\frac {2 \cos (a+b x) \csc ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \sqrt {\csc (a+b x)} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 50, normalized size = 0.75 \[ -\frac {2 \csc ^{\frac {3}{2}}(a+b x) \left (\cos (a+b x)+\sin ^{\frac {3}{2}}(a+b x) F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^(5/2),x]

[Out]

(-2*Csc[a + b*x]^(3/2)*(Cos[a + b*x] + EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2)))/(3*b)

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\csc \left (b x + a\right )^{\frac {5}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^(5/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^(5/2), x)

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maple [A]  time = 2.43, size = 88, normalized size = 1.31 \[ \frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )^{\frac {3}{2}} \cos \left (b x +a \right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^(5/2),x)

[Out]

1/3/sin(b*x+a)^(3/2)*((sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF((sin(b*x+a)+1
)^(1/2),1/2*2^(1/2))*sin(b*x+a)-2*cos(b*x+a)^2)/cos(b*x+a)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(a + b*x))^(5/2),x)

[Out]

int((1/sin(a + b*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**(5/2),x)

[Out]

Integral(csc(a + b*x)**(5/2), x)

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