Integrand size = 14, antiderivative size = 62 \[ \int \sqrt {a+b \sin (e+f x)} \, dx=\frac {2 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \] Output:
-2*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(f *x+e))^(1/2)/f/((a+b*sin(f*x+e))/(a+b))^(1/2)
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \sqrt {a+b \sin (e+f x)} \, dx=-\frac {2 E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \] Input:
Integrate[Sqrt[a + b*Sin[e + f*x]],x]
Output:
(-2*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x ]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)])
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3134, 3042, 3132}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a+b \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a+b \sin (e+f x)}dx\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\) |
Input:
Int[Sqrt[a + b*Sin[e + f*x]],x]
Output:
(2*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/ (f*Sqrt[(a + b*Sin[e + f*x])/(a + b)])
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs. \(2(61)=122\).
Time = 1.34 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.85
method | result | size |
default | \(\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {b \left (-1+\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \left (a \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b -\operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a -\operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b \right )}{b \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\) | \(239\) |
risch | \(\text {Expression too large to display}\) | \(1514\) |
Input:
int((a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-b*(-1+sin(f*x+e))/(a+b))^(1/2)*(- b*(1+sin(f*x+e))/(a-b))^(1/2)/b*(a*EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2 ),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+ b))^(1/2))*b-EllipticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)) *a-EllipticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b)/cos(f* x+e)/(a+b*sin(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 351, normalized size of antiderivative = 5.66 \[ \int \sqrt {a+b \sin (e+f x)} \, dx=\frac {2 \, {\left (a \sqrt {\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) + a \sqrt {-\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right ) - 3 i \, \sqrt {\frac {1}{2} i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 i \, \sqrt {-\frac {1}{2} i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )\right )\right )}}{3 \, b f} \] Input:
integrate((a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
2/3*(a*sqrt(1/2*I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*( 8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) - 3*I*b*sin(f*x + e) - 2*I *a)/b) + a*sqrt(-1/2*I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8 /27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) + 3*I*b*sin(f*x + e) + 2*I*a)/b) - 3*I*sqrt(1/2*I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^ 2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^ 2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) - 3*I*b*sin (f*x + e) - 2*I*a)/b)) + 3*I*sqrt(-1/2*I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*( 4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e ) + 3*I*b*sin(f*x + e) + 2*I*a)/b)))/(b*f)
\[ \int \sqrt {a+b \sin (e+f x)} \, dx=\int \sqrt {a + b \sin {\left (e + f x \right )}}\, dx \] Input:
integrate((a+b*sin(f*x+e))**(1/2),x)
Output:
Integral(sqrt(a + b*sin(e + f*x)), x)
\[ \int \sqrt {a+b \sin (e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*sin(f*x + e) + a), x)
\[ \int \sqrt {a+b \sin (e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*sin(f*x + e) + a), x)
Time = 16.42 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+b \sin (e+f x)} \, dx=\frac {2\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |\frac {2\,b}{a+b}\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}}{f\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}} \] Input:
int((a + b*sin(e + f*x))^(1/2),x)
Output:
(2*ellipticE(e/2 - pi/4 + (f*x)/2, (2*b)/(a + b))*(a + b*sin(e + f*x))^(1/ 2))/(f*((a + b*sin(e + f*x))/(a + b))^(1/2))
\[ \int \sqrt {a+b \sin (e+f x)} \, dx=\int \sqrt {\sin \left (f x +e \right ) b +a}d x \] Input:
int((a+b*sin(f*x+e))^(1/2),x)
Output:
int(sqrt(sin(e + f*x)*b + a),x)