\(\int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx\) [238]

Optimal result

Integrand size = 25, antiderivative size = 74 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))} \]

[Out]

-2*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))-2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1
/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/a/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2762, 2721, 2719} \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}} \]

[In]

Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]

[Out]

(-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]))

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]

Rule 2762

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*Sin[e + f*x]))), x] + Dist[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] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))}-\frac {\int \sqrt {e \cos (c+d x)} \, dx}{a} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{a \sqrt {\cos (c+d x)}} \\ & = -\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))} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=-\frac {2^{3/4} (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 a d e (1+\sin (c+d x))^{3/4}} \]

[In]

Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]

[Out]

-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))

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {2 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}{\sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a d}\)	\(118\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\frac {{\left (-i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2} \sin \left (d x + c\right ) - i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + {\left (i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2} \sin \left (d x + c\right ) + i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]

[In]

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

[Out]

((-I*sqrt(2)*cos(d*x + c) - I*sqrt(2)*sin(d*x + c) - I*sqrt(2))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInv
erse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + (I*sqrt(2)*cos(d*x + c) + I*sqrt(2)*sin(d*x + c) + I*sqrt(2))*sq
rt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*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)

Sympy [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate((e*cos(d*x+c))**(1/2)/(a+a*sin(d*x+c)),x)

[Out]

Integral(sqrt(e*cos(c + d*x))/(sin(c + d*x) + 1), x)/a

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(e*cos(d*x + c))/(a*sin(d*x + c) + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(e*cos(d*x + c))/(a*sin(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{a+a\,\sin \left (c+d\,x\right )} \,d x \]

[In]

int((e*cos(c + d*x))^(1/2)/(a + a*sin(c + d*x)),x)

[Out]

int((e*cos(c + d*x))^(1/2)/(a + a*sin(c + d*x)), x)