\(\int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx\) [574]

Optimal result

Integrand size = 29, antiderivative size = 105 \[ \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {\sqrt {a} (c+d) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.98 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.26 \[ \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {a (1+\sin (e+f x))} \left (\frac {i (c+d) e^{\frac {1}{2} i (e+f x)} \sqrt {2 c-i d e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \left ((-1)^{3/4} \sqrt {2} \arctan \left (\frac {\sqrt [4]{-1} \left (d-i c e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )-(1+i) \text {arctanh}\left (\frac {(-1)^{3/4} \left (c-i d e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )} f}-\frac {(2-2 i) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}{f}\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )} \] Input:

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

Output:

((1/4 + I/4)*Sqrt[a*(1 + Sin[e + f*x])]*((I*(c + d)*E^((I/2)*(e + f*x))*Sq 
rt[2*c - (I*d*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*((-1)^(3/4)*Sqr 
t[2]*ArcTan[((-1)^(1/4)*(d - I*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 + I)*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[d]*Sqrt[-2*c*E^(I*(e + f*x)) + I*d*(-1 
 + E^((2*I)*(e + f*x)))]*f) - ((2 - 2*I)*(Cos[(e + f*x)/2] - Sin[(e + f*x) 
/2])*Sqrt[c + d*Sin[e + f*x]])/f))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3249, 3042, 3254, 218}

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.

\(\displaystyle \int \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}dx\)

\(\Big \downarrow \) 3249

\(\displaystyle \frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 3254

\(\displaystyle -\frac {a (c+d) \int \frac {1}{\frac {a^2 d \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}+a}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\sqrt {a} (c+d) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3249

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) 
^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 
2*n + 1)))   Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), 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] && GtQ[n, 0] && IntegerQ[2*n]
 

rule 3254

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]], x_Symbol] :> Simp[-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]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (89) = 178\).

Time = 0.35 (sec) , antiderivative size = 945, normalized size of antiderivative = 9.00 \[ \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*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)) - 8*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos 
(f*x + e) - sin(f*x + e) + 1))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/4* 
(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*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*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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