3.87 \(\int \frac{\sin ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{\text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{6 b}+\frac{\sin ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]

[Out]

EllipticF[a - Pi/4 + b*x, 2]/(6*b) + Sin[a + b*x]^2/(3*b*Sin[2*a + 2*b*x]^(3/2))

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Rubi [A]  time = 0.0376517, antiderivative size = 48, 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 = {4296, 2641} \[ \frac{\sin ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

EllipticF[a - Pi/4 + b*x, 2]/(6*b) + Sin[a + b*x]^2/(3*b*Sin[2*a + 2*b*x]^(3/2))

Rule 4296

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*(p + 1)), x] + Dist[(e^2*(m + 2*p + 2))/(4*g^2*(p + 1)), Int[(e*Sin[a
+ b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[
d/b, 2] &&  !IntegerQ[p] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && Int
egersQ[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)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=\frac{\sin ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{1}{6} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{F\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{6 b}+\frac{\sin ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}\\ \end{align*}

Mathematica [A]  time = 0.190226, size = 83, normalized size = 1.73 \[ \frac{\sqrt{\sin (2 (a+b x))} \sec ^2(a+b x)-\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) \text{EllipticF}\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x)),\frac{1}{2}\right )}{\sqrt{\sin (2 (a+b x))+1}}}{12 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

(Sec[a + b*x]^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)]])/(12*b)

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Maple [A]  time = 11.461, size = 123, normalized size = 2.6 \begin{align*}{\frac{1}{12\,\cos \left ( 2\,bx+2\,a \right ) b} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( 2\,bx+2\,a \right ) -2\, \left ( \cos \left ( 2\,bx+2\,a \right ) \right ) ^{2}+2\,\cos \left ( 2\,bx+2\,a \right ) \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

1/12/sin(2*b*x+2*a)^(3/2)/cos(2*b*x+2*a)*((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))*sin(2*b*x+2*a)-2*cos(2*b*x+2*a)^2+2*cos(2*b*x+2*a))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sin(b*x + a)^2/sin(2*b*x + 2*a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x + a\right )^{2} - 1}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \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)^(5/2),x, algorithm="fricas")

[Out]

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

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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)**(5/2),x)

[Out]

Timed out

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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)^(5/2),x, algorithm="giac")

[Out]

Timed out