\(\int \sqrt {c \sin (a+b x)} \, dx\) [28]

Optimal result

Integrand size = 12, antiderivative size = 43 \[ \int \sqrt {c \sin (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2721, 2719} \[ \int \sqrt {c \sin (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (a+b x)}} \]

[In]

Int[Sqrt[c*Sin[a + b*x]],x]

[Out]

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

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 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c \sin (a+b x)} \int \sqrt {\sin (a+b x)} \, dx}{\sqrt {\sin (a+b x)}} \\ & = \frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \sqrt {c \sin (a+b x)} \, dx=-\frac {2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (a+b x)}} \]

[In]

Integrate[Sqrt[c*Sin[a + b*x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.28

method	result	size
default	\(-\frac {c \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \left (2 E\left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-F\left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b}\)	\(98\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {-i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}+\frac {i \left (\frac {2 i \left (-i c \,{\mathrm e}^{2 i \left (b x +a \right )}+i c \right )}{c \sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (-i c \,{\mathrm e}^{2 i \left (b x +a \right )}+i c \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (b x +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 E\left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-i c \,{\mathrm e}^{3 i \left (b x +a \right )}+i c \,{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {-i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {-i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\)	\(297\)

[In]

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

[Out]

-c*(-sin(b*x+a)+1)^(1/2)*(2*sin(b*x+a)+2)^(1/2)*sin(b*x+a)^(1/2)*(2*EllipticE((-sin(b*x+a)+1)^(1/2),1/2*2^(1/2
))-EllipticF((-sin(b*x+a)+1)^(1/2),1/2*2^(1/2)))/cos(b*x+a)/(c*sin(b*x+a))^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.56 \[ \int \sqrt {c \sin (a+b x)} \, dx=\frac {i \, \sqrt {2} \sqrt {-i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]

[In]

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

[Out]

(I*sqrt(2)*sqrt(-I*c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) - I*sqrt
(2)*sqrt(I*c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a))))/b

Sympy [F]

\[ \int \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {c \sin {\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(c*sin(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)), x)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \sqrt {c \sin (a+b x)} \, dx=\frac {2\,\sqrt {c\,\sin \left (a+b\,x\right )}\,\mathrm {E}\left (\frac {a}{2}-\frac {\pi }{4}+\frac {b\,x}{2}\middle |2\right )}{b\,\sqrt {\sin \left (a+b\,x\right )}} \]

[In]

int((c*sin(a + b*x))^(1/2),x)

[Out]

(2*(c*sin(a + b*x))^(1/2)*ellipticE(a/2 - pi/4 + (b*x)/2, 2))/(b*sin(a + b*x)^(1/2))