\(\int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\) [85]

Optimal result

Integrand size = 22, antiderivative size = 40 \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\frac {\operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4383, 2720} \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\frac {\operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \]

[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 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 4383

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
*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] && E
qQ[d/b, 2] &&  !IntegerQ[p] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\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 {\operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{2 b}-\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.88 \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=-\frac {2 \sqrt {\sin (2 (a+b x))}+\frac {\sqrt {2} \operatorname {EllipticF}\left (\arcsin (\cos (a+b x)-\sin (a+b x)),\frac {1}{2}\right ) (\cos (a+b x)+\sin (a+b x))}{\sqrt {1+\sin (2 (a+b x))}}}{4 b} \]

[In]

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

[Out]

-1/4*(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)]])/b

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 14.82 (sec) , antiderivative size = 61245868, normalized size of antiderivative = 1531146.70

method	result	size
default	\(\text {Expression too large to display}\)	\(61245868\)

[In]

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

[Out]

result too large to display

Fricas [F]

\[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}} \,d x } \]

[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)]

Timed out. \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{\sqrt {\sin \left (2\,a+2\,b\,x\right )}} \,d x \]

[In]

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

[Out]

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