Integrand size = 25, antiderivative size = 74 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))} \] Output:
-2*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a/d/cos(d*x+ c)^(1/2)-2*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2^{3/4} (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 a d e (1+\sin (c+d x))^{3/4}} \] Input:
Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]
Output:
-1/3*(2^(3/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 5/4, 7/4, (1 - Sin[c + d*x])/2])/(a*d*e*(1 + Sin[c + d*x])^(3/4))
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3162, 3042, 3121, 3042, 3119}
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.
\(\displaystyle \int \frac {\sqrt {e \cos (c+d x)}}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {e \cos (c+d x)}}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3162 |
\(\displaystyle -\frac {\int \sqrt {e \cos (c+d x)}dx}{a}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle -\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}}\) |
Input:
Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]
Output:
(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x] ]) - (2*(e*Cos[c + d*x])^(3/2))/(d*e*(a + a*Sin[c + d*x]))
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]
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)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59
method | result | size |
default | \(\frac {2 \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}{\sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a d}\) | \(118\) |
Input:
int((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
2/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)/a*(2*sin(1/2*d*x+ 1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/ 2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-sin(1/2*d*x+1/2*c))* e/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} \sqrt {e} {\left (i \, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {\frac {1}{2}} \sqrt {e} {\left (-i \, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - i\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {e \cos \left (d x + c\right )} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}\right )}}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \] Input:
integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-2*(sqrt(1/2)*sqrt(e)*(I*cos(d*x + c) + I*sin(d*x + c) + I)*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + sqrt (1/2)*sqrt(e)*(-I*cos(d*x + c) - I*sin(d*x + c) - I)*weierstrassZeta(-4, 0 , weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + sqrt(e*cos( d*x + c))*(cos(d*x + c) - sin(d*x + c) + 1))/(a*d*cos(d*x + c) + a*d*sin(d *x + c) + a*d)
\[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate((e*cos(d*x+c))**(1/2)/(a+a*sin(d*x+c)),x)
Output:
Integral(sqrt(e*cos(c + d*x))/(sin(c + d*x) + 1), x)/a
\[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{a \sin \left (d x + c\right ) + a} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
integrate(sqrt(e*cos(d*x + c))/(a*sin(d*x + c) + a), x)
\[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{a \sin \left (d x + c\right ) + a} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
integrate(sqrt(e*cos(d*x + c))/(a*sin(d*x + c) + a), x)
Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{a+a\,\sin \left (c+d\,x\right )} \,d x \] Input:
int((e*cos(c + d*x))^(1/2)/(a + a*sin(c + d*x)),x)
Output:
int((e*cos(c + d*x))^(1/2)/(a + a*sin(c + d*x)), x)
\[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\sin \left (d x +c \right )+1}d x \right )}{a} \] Input:
int((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c)),x)
Output:
(sqrt(e)*int(sqrt(cos(c + d*x))/(sin(c + d*x) + 1),x))/a