\(\int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx\) [300]

Optimal result

Integrand size = 27, antiderivative size = 169 \[ \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))} \]

[Out]

2*arcsinh((e*cos(d*x+c))^(1/2)/e^(1/2))*e^(1/2)*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(a+a*cos(d*x+c)+
a*sin(d*x+c))+2*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+cos(d*x+c))^(1/2))*e^(1/2)*(1+cos(d*x+c))^(1
/2)*(a+a*sin(d*x+c))^(1/2)/d/(a+a*cos(d*x+c)+a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2763, 2854, 209, 2912, 65, 221} \[ \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)} \]

[In]

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

[Out]

(2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Co
s[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c
+ d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2763

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[g*Sqrt[1
+ Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])), Int[Sqrt[1 + Cos[e + f*x]]/
Sqrt[g*Cos[e + f*x]], x], x] - Dist[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a
*Sin[e + f*x])), Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f,
g}, x] && EqQ[a^2 - b^2, 0]

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{a+a \cos (c+d x)+a \sin (c+d x)}-\frac {\left (e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{a+a \cos (c+d x)+a \sin (c+d x)} \\ & = \frac {\left (e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = \frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = \frac {2 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \sqrt [4]{2} (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e \sqrt [4]{1+\sin (c+d x)} \sqrt {a (1+\sin (c+d x))}} \]

[In]

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

[Out]

(-2*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c +
d*x])^(1/4)*Sqrt[a*(1 + Sin[c + d*x])])

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\sqrt {e \cos \left (d x +c \right )}\, \left (\arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )\right ) \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\)	\(133\)

[In]

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

[Out]

1/d*(e*cos(d*x+c))^(1/2)*(arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)))*(1+cos(d*x+c)+sin(d*x+c))/(1+cos(d*x+c))/(a*(1+sin(d*x+c)))^(1/2)/(-cos(d*x+c)/(
1+cos(d*x+c)))^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 877, normalized size of antiderivative = 5.19 \[ \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(-e^2/(a^2*d^4))^(1/4)*log((2*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*(e*sin(d*x + c) + (a*d^2*cos(d
*x + c) + a*d^2)*sqrt(-e^2/(a^2*d^4))) + (2*a^2*d^3*cos(d*x + c)^2 + a^2*d^3*cos(d*x + c) - a^2*d^3*sin(d*x +
c) - a^2*d^3)*(-e^2/(a^2*d^4))^(3/4) + (a*d*e*cos(d*x + c) + a*d*e + (2*a*d*e*cos(d*x + c) + a*d*e)*sin(d*x +
c))*(-e^2/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 1/2*(-e^2/(a^2*d^4))^(1/4)*log((2*sqrt(e*cos(
d*x + c))*sqrt(a*sin(d*x + c) + a)*(e*sin(d*x + c) + (a*d^2*cos(d*x + c) + a*d^2)*sqrt(-e^2/(a^2*d^4))) - (2*a
^2*d^3*cos(d*x + c)^2 + a^2*d^3*cos(d*x + c) - a^2*d^3*sin(d*x + c) - a^2*d^3)*(-e^2/(a^2*d^4))^(3/4) - (a*d*e
*cos(d*x + c) + a*d*e + (2*a*d*e*cos(d*x + c) + a*d*e)*sin(d*x + c))*(-e^2/(a^2*d^4))^(1/4))/(cos(d*x + c) + s
in(d*x + c) + 1)) - 1/2*I*(-e^2/(a^2*d^4))^(1/4)*log((2*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*(e*sin(d
*x + c) - (a*d^2*cos(d*x + c) + a*d^2)*sqrt(-e^2/(a^2*d^4))) + (2*I*a^2*d^3*cos(d*x + c)^2 + I*a^2*d^3*cos(d*x
+ c) - I*a^2*d^3*sin(d*x + c) - I*a^2*d^3)*(-e^2/(a^2*d^4))^(3/4) + (-I*a*d*e*cos(d*x + c) - I*a*d*e + (-2*I*
a*d*e*cos(d*x + c) - I*a*d*e)*sin(d*x + c))*(-e^2/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) + 1/2*I
*(-e^2/(a^2*d^4))^(1/4)*log((2*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*(e*sin(d*x + c) - (a*d^2*cos(d*x
+ c) + a*d^2)*sqrt(-e^2/(a^2*d^4))) + (-2*I*a^2*d^3*cos(d*x + c)^2 - I*a^2*d^3*cos(d*x + c) + I*a^2*d^3*sin(d*
x + c) + I*a^2*d^3)*(-e^2/(a^2*d^4))^(3/4) + (I*a*d*e*cos(d*x + c) + I*a*d*e + (2*I*a*d*e*cos(d*x + c) + I*a*d
*e)*sin(d*x + c))*(-e^2/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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