3.1.14 \(\int \frac {1}{\sin ^{\frac {3}{2}}(b x)} \, dx\) [14]

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 = {2716, 2719} \begin {gather*} \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 {gather*}

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 2716

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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], 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 = 0.04, size = 32, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

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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Maple [A]
time = 0.05, size = 110, normalized size = 2.97

method	result	size
default	\(\frac {2 \sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticE \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x \right )\right )}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 81, normalized size = 2.19 \begin {gather*} \frac {-i \, \sqrt {2} \sqrt {-i} \sin \left (b x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right )\right ) + i \, \sqrt {2} \sqrt {i} \sin \left (b x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) - i \, \sin \left (b x\right )\right )\right ) - 2 \, \cos \left (b x\right ) \sqrt {\sin \left (b x\right )}}{b \sin \left (b x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*sqrt(2)*sqrt(-I)*sin(b*x)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x) + I*sin(b*x))) + I*sqrt
(2)*sqrt(I)*sin(b*x)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x) - I*sin(b*x))) - 2*cos(b*x)*sqrt
(sin(b*x)))/(b*sin(b*x))

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

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

[Out]

integrate(sin(b*x)^(-3/2), x)

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Mupad [B]
time = 0.47, size = 34, normalized size = 0.92 \begin {gather*} -\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 )}} \end {gather*}

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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