3.85 \(\int \frac{\sin ^2(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{2 b}-\frac{\sqrt{\sin (2 a+2 b x)}}{2 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0369828, antiderivative size = 40, normalized size of antiderivative = 1., 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 = {4298, 2641} \[ \frac{F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b}-\frac{\sqrt{\sin (2 a+2 b x)}}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4298

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> -Simp[(e^2*(e*Sin[
a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + 2*p)), x] + Dist[(e^2*(m + p - 1))/(m + 2*p), Int[(e*S
in[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] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]

Rule 2641

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

Rubi steps

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

Mathematica [A]  time = 0.244552, size = 75, normalized size = 1.88 \[ -\frac{\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) \text{EllipticF}\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x)),\frac{1}{2}\right )}{\sqrt{\sin (2 (a+b x))+1}}+2 \sqrt{\sin (2 (a+b x))}}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[Sin[2*(a + b*x)]] + (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[
a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/(4*b)

________________________________________________________________________________________

Maple [B]  time = 8.632, size = 53350427, normalized size = 1333760.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{2}}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (b x + a\right )^{2} - 1}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out