3.2.9 \(\int \csc ^2(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\) [109]

Optimal. Leaf size=44 \[ -\frac {2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b}-\frac {\csc ^2(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 44, 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, 2719} \begin {gather*} -\frac {2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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

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Mathematica [A]
time = 0.14, size = 37, normalized size = 0.84 \begin {gather*} -\frac {2 \left (E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\cot (a+b x) \sqrt {\sin (2 (a+b x))}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(175\) vs. \(2(67)=134\).
time = 10.64, size = 176, normalized size = 4.00

method	result	size
default	\(\frac {2 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \EllipticE \left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \EllipticF \left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (2 x b +2 a \right )\right )-2 \cos \left (2 x b +2 a \right )}{\cos \left (2 x b +2 a \right ) \sqrt {\sin \left (2 x b +2 a \right )}\, b}\)	\(176\)

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,method=_RETURNVERBOSE)

[Out]

1/cos(2*b*x+2*a)/sin(2*b*x+2*a)^(1/2)*(2*(sin(2*b*x+2*a)+1)^(1/2)*(-2*sin(2*b*x+2*a)+2)^(1/2)*(-sin(2*b*x+2*a)
)^(1/2)*EllipticE((sin(2*b*x+2*a)+1)^(1/2),1/2*2^(1/2))-(sin(2*b*x+2*a)+1)^(1/2)*(-2*sin(2*b*x+2*a)+2)^(1/2)*(
-sin(2*b*x+2*a))^(1/2)*EllipticF((sin(2*b*x+2*a)+1)^(1/2),1/2*2^(1/2))-2*cos(2*b*x+2*a)^2-2*cos(2*b*x+2*a))/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(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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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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 {\sqrt {\sin \left (2\,a+2\,b\,x\right )}}{{\sin \left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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