3.14 \(\int \frac {1}{\sin ^{\frac {3}{2}}(b x)} \, dx\)

Optimal. Leaf size=37 \[ \frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b}-\frac {2 \cos (b x)}{b \sqrt {\sin (b x)}} \]

[Out]

2*(sin(1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/4*Pi+1/2*b*x)*EllipticE(cos(1/4*Pi+1/2*b*x),2^(1/2))/b-2*cos(b*x)/b/sin(
b*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2636, 2639} \[ \frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b}-\frac {2 \cos (b x)}{b \sqrt {\sin (b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[b*x]^(-3/2),x]

[Out]

(2*EllipticE[Pi/4 - (b*x)/2, 2])/b - (2*Cos[b*x])/(b*Sqrt[Sin[b*x]])

Rule 2636

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)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2639

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

Rubi steps

\begin {align*} \int \frac {1}{\sin ^{\frac {3}{2}}(b x)} \, dx &=-\frac {2 \cos (b x)}{b \sqrt {\sin (b x)}}-\int \sqrt {\sin (b x)} \, dx\\ &=\frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b}-\frac {2 \cos (b x)}{b \sqrt {\sin (b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 32, normalized size = 0.86 \[ \frac {2 \left (E\left (\left .\frac {1}{4} (\pi -2 b x)\right |2\right )-\frac {\cos (b x)}{\sqrt {\sin (b x)}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[b*x]^(-3/2),x]

[Out]

(2*(EllipticE[(Pi - 2*b*x)/4, 2] - Cos[b*x]/Sqrt[Sin[b*x]]))/b

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {\sin \left (b x\right )}}{\cos \left (b x\right )^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(3/2),x, algorithm="fricas")

[Out]

integral(-sqrt(sin(b*x))/(cos(b*x)^2 - 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sin \left (b x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sin(b*x)^(-3/2), x)

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maple [A]  time = 0.07, size = 110, normalized size = 2.97 \[ \frac {2 \sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticE \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x \right )\right )}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x)^(3/2),x)

[Out]

(2*(sin(b*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*EllipticE((sin(b*x)+1)^(1/2),1/2*2^(1/2))-(sin(b
*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*EllipticF((sin(b*x)+1)^(1/2),1/2*2^(1/2))-2*cos(b*x)^2)/c
os(b*x)/sin(b*x)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sin \left (b x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x)^(-3/2), x)

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mupad [B]  time = 0.47, size = 34, normalized size = 0.92 \[ -\frac {\cos \left (b\,x\right )\,{\left ({\sin \left (b\,x\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (b\,x\right )}^2\right )}{b\,\sqrt {\sin \left (b\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(b*x)^(3/2),x)

[Out]

-(cos(b*x)*(sin(b*x)^2)^(1/4)*hypergeom([1/2, 5/4], 3/2, cos(b*x)^2))/(b*sin(b*x)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sin ^{\frac {3}{2}}{\left (b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(b*x)**(3/2),x)

[Out]

Integral(sin(b*x)**(-3/2), x)

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