\(\int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx\) [567]

Optimal result

Integrand size = 29, antiderivative size = 61 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2854, 211} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f} \]

[In]

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

[Out]

(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[
d]*f)

Rule 211

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

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 290, normalized size of antiderivative = 4.75 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{\frac {1}{2} i (e+f x)} \sqrt {6 c-3 i d e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \left ((1+i) \arctan \left (\frac {(-1)^{3/4} \left (i d+c e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )-(-1)^{3/4} \sqrt {2} \text {arctanh}\left (\frac {(-1)^{3/4} \left (c-i d e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )\right ) \sqrt {1+\sin (e+f x)}}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[Sqrt[3 + 3*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]],x]

[Out]

((1/2 + I/2)*E^((I/2)*(e + f*x))*Sqrt[6*c - ((3*I)*d*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*((1 + I)*Arc
Tan[((-1)^(3/4)*(I*d + c*E^(I*(e + f*x))))/(Sqrt[d]*Sqrt[-2*c*E^(I*(e + f*x)) + I*d*(-1 + E^((2*I)*(e + f*x)))
])] - (-1)^(3/4)*Sqrt[2]*ArcTanh[((-1)^(3/4)*(c - I*d*E^(I*(e + f*x))))/(Sqrt[d]*Sqrt[-2*c*E^(I*(e + f*x)) + I
*d*(-1 + E^((2*I)*(e + f*x)))])])*Sqrt[1 + Sin[e + f*x]])/(Sqrt[d]*Sqrt[-2*c*E^(I*(e + f*x)) + I*d*(-1 + E^((2
*I)*(e + f*x)))]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1140\) vs. \(2(49)=98\).

Time = 1.75 (sec) , antiderivative size = 1141, normalized size of antiderivative = 18.70

\[\text {Expression too large to display}\]

[In]

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

[Out]

1/f*(a*(sin(f*x+e)+1))^(1/2)*(cos(f*x+e)^2+sin(f*x+e)^2-2*cos(f*x+e)+1)*(c+d*sin(f*x+e))^(1/2)*((c+d*sin(f*x+e
))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/
2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(d^2/c^2)^(1/2)*c^3*d-2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e
)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(d^2/c^2)^(1/2)*c^2*d^2+arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*
sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(d^2/c^2)^(1/2)*c*d^3-(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2
/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^
2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*
(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*(d^2/c^2)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6
*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*c-arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)
^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*c^2*d^2+2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2
)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*c*d^3-arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f
*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*d^4-(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e
)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d
*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2
+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^
(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*d)*(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/(cos(f*x+e)-1-sin(f*x+e))/(c
*cos(f*x+e)^2-2*d*sin(f*x+e)*cos(f*x+e)+c*sin(f*x+e)^2-2*c*cos(f*x+e)+2*d*sin(f*x+e)+c)/d^3/(-(d^2/c^2)^(1/2)*
c)^(1/2)/(c^2-2*c*d+d^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (49) = 98\).

Time = 0.53 (sec) , antiderivative size = 777, normalized size of antiderivative = 12.74 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\left [\frac {\sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right )}{4 \, f}, \frac {\sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \]

[In]

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

[Out]

[1/4*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a
*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c
^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d
^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 +
31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^
3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e)
+ a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(
f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*
d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a
*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1))/f, 1/2*sqrt(a/d)*arctan(1/4*(
8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f
*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d
+ 2*a*d^2)*cos(f*x + e)))/f]

Sympy [F]

\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

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

[Out]

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