3.2.10 \(\int \frac {\csc ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\) [110]

Optimal. Leaf size=48 \[ \frac {2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{3 b}-\frac {\csc ^2(a+b x) \sqrt {\sin (2 a+2 b x)}}{3 b} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4385, 2720} \begin {gather*} \frac {2 F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b}-\frac {\sqrt {\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2/Sqrt[Sin[2*a + 2*b*x]],x]

[Out]

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

Rule 2720

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

Rule 4385

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Sin[a + b*
x])^m*((g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1))), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a
+ b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b,
2] &&  !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rubi steps

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

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Mathematica [A]
time = 1.03, size = 82, normalized size = 1.71 \begin {gather*} -\frac {\csc ^2(a+b x) \sqrt {\sin (2 (a+b x))}+\frac {\sqrt {2} F\left (\text {ArcSin}(\cos (a+b x)-\sin (a+b x))\left |\frac {1}{2}\right .\right ) (\cos (a+b x)+\sin (a+b x))}{\sqrt {1+\sin (2 (a+b x))}}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2/Sqrt[Sin[2*a + 2*b*x]],x]

[Out]

-1/3*(Csc[a + b*x]^2*Sqrt[Sin[2*(a + b*x)]] + (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Co
s[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/b

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{2}\left (x b +a \right )}{\sqrt {\sin \left (2 x b +2 a \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(csc(b*x + a)^2/sqrt(sin(2*b*x + 2*a)), x)

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Fricas [C] Result contains complex when optimal does not.
time = 1.22, size = 101, normalized size = 2.10 \begin {gather*} -\frac {\sqrt {2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

integrate(csc(b*x + a)^2/sqrt(sin(2*b*x + 2*a)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,\sqrt {\sin \left (2\,a+2\,b\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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