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

Optimal. Leaf size=43 \[ -\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b}-\frac {2 \cos (a+b x)}{b \sqrt {\sin (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2636, 2639} \[ -\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b}-\frac {2 \cos (a+b x)}{b \sqrt {\sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^(-3/2),x]

[Out]

(-2*EllipticE[(a - Pi/2 + b*x)/2, 2])/b - (2*Cos[a + b*x])/(b*Sqrt[Sin[a + 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}}(a+b x)} \, dx &=-\frac {2 \cos (a+b x)}{b \sqrt {\sin (a+b x)}}-\int \sqrt {\sin (a+b x)} \, dx\\ &=-\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b}-\frac {2 \cos (a+b x)}{b \sqrt {\sin (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 39, normalized size = 0.91 \[ \frac {2 \left (E\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )-\frac {\cos (a+b x)}{\sqrt {\sin (a+b x)}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^(-3/2),x]

[Out]

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

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {\sin \left (b x + a\right )}}{\cos \left (b x + a\right )^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(-3/2), x)

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maple [A]  time = 0.04, size = 132, normalized size = 3.07 \[ \frac {2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x +a \right )\right )}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x + a)^(-3/2), x)

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mupad [B]  time = 0.51, size = 42, normalized size = 0.98 \[ -\frac {\cos \left (a+b\,x\right )\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,\sqrt {\sin \left (a+b\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*x)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x)**(-3/2), x)

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