Integrand size = 23, antiderivative size = 66 \[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e} \]
-2*a*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*Ellipti cF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/d/(e*sin(d*x+c))^(1 /2)+2*b*(e*sin(d*x+c))^(1/2)/d/e
Time = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {2 \left (-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )}{d \sqrt {e \sin (c+d x)}} \]
(2*(-(a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]) + b*Sin[c + d*x]))/(d*Sqrt[e*Sin[c + d*x]])
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3148, 3042, 3121, 3042, 3120}
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.
\(\displaystyle \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-b \sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3148 |
\(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \frac {1}{\sqrt {e \sin (c+d x)}}dx+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 a \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\) |
(2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*Sqrt[e*Sin[c + d*x]])/(d*e)
3.1.37.3.1 Defintions of rubi rules used
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Time = 2.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {a \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right ) \cos \left (d x +c \right ) b}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(92\) |
parts | \(-\frac {a \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 b \sqrt {e \sin \left (d x +c \right )}}{d e}\) | \(95\) |
risch | \(-\frac {i b \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {-i e \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) {\mathrm e}^{-i \left (d x +c \right )}}}-\frac {i a \sqrt {{\mathrm e}^{i \left (d x +c \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (d x +c \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (d x +c \right )}}\, F\left (\sqrt {{\mathrm e}^{i \left (d x +c \right )}+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-i e \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) {\mathrm e}^{i \left (d x +c \right )}}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {-i e \,{\mathrm e}^{3 i \left (d x +c \right )}+i e \,{\mathrm e}^{i \left (d x +c \right )}}\, \sqrt {-i e \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) {\mathrm e}^{-i \left (d x +c \right )}}}\) | \(236\) |
-1/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*(a*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2 )^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-2*sin (d*x+c)*cos(d*x+c)*b)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {\sqrt {2} a \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} a \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {e \sin \left (d x + c\right )} b}{d e} \]
(sqrt(2)*a*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + sqrt(2)*a*sqrt(I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin( d*x + c)) + 2*sqrt(e*sin(d*x + c))*b)/(d*e)
\[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int \frac {a + b \cos {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]
\[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]
Time = 13.74 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=-\frac {2\,\sqrt {\sin \left (c+d\,x\right )}\,\left (a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {c}{2}-\frac {d\,x}{2}\middle |2\right )-b\,\sqrt {\sin \left (c+d\,x\right )}\right )}{d\,\sqrt {e\,\sin \left (c+d\,x\right )}} \]