Integrand size = 39, antiderivative size = 149 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{d f}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{d \sqrt {c+d} f} \]
-2*arctan(cos(f*x+e)*a^(1/2)*g^(1/2)/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e)) ^(1/2))*a^(1/2)*g^(1/2)/d/f+2*arctan(cos(f*x+e)*a^(1/2)*c^(1/2)*g^(1/2)/(c +d)^(1/2)/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2))*a^(1/2)*c^(1/2)*g^( 1/2)/d/f/(c+d)^(1/2)
Result contains complex when optimal does not.
Time = 3.40 (sec) , antiderivative size = 662, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {5}{2} i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )^{5/2} \left (-2 i \sqrt {-1+e^{2 i (e+f x)}}+\left (i+\frac {c-d}{\sqrt {-c^2+d^2}}\right ) \sqrt {-1+e^{2 i (e+f x)}}+\left (i+\frac {-c+d}{\sqrt {-c^2+d^2}}\right ) \sqrt {-1+e^{2 i (e+f x)}}+2 i \arctan \left (\sqrt {-1+e^{2 i (e+f x)}}\right )+\frac {\left (i+\frac {-c+d}{\sqrt {-c^2+d^2}}\right ) \left (\sqrt {2} \sqrt {c} \sqrt {c+i \sqrt {-c^2+d^2}} \arctan \left (\frac {d-\left (-i c+\sqrt {-c^2+d^2}\right ) e^{i (e+f x)}}{\sqrt {2} \sqrt {c} \sqrt {c+i \sqrt {-c^2+d^2}} \sqrt {-1+e^{2 i (e+f x)}}}\right )+\left (-i c+\sqrt {-c^2+d^2}\right ) \text {arctanh}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right )}{d}+\frac {\left (i+\frac {c-d}{\sqrt {-c^2+d^2}}\right ) \left (\sqrt {2} \sqrt {c} \sqrt {c-i \sqrt {-c^2+d^2}} \arctan \left (\frac {d+\left (i c+\sqrt {-c^2+d^2}\right ) e^{i (e+f x)}}{\sqrt {2} \sqrt {c} \sqrt {c-i \sqrt {-c^2+d^2}} \sqrt {-1+e^{2 i (e+f x)}}}\right )-\left (i c+\sqrt {-c^2+d^2}\right ) \text {arctanh}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right )}{d}\right ) \sqrt {g \sin (e+f x)} \sqrt {a (1+\sin (e+f x))}}{\sqrt {2} d \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{5/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\sin (e+f x)}} \]
((1/2 + I/2)*(-1 + E^((2*I)*(e + f*x)))^(5/2)*((-2*I)*Sqrt[-1 + E^((2*I)*( e + f*x))] + (I + (c - d)/Sqrt[-c^2 + d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] + (I + (-c + d)/Sqrt[-c^2 + d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] + (2*I)* ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] + ((I + (-c + d)/Sqrt[-c^2 + d^2])* (Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]]*ArcTan[(d - ((-I)*c + Sqrt[- c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]] *Sqrt[-1 + E^((2*I)*(e + f*x))])] + ((-I)*c + Sqrt[-c^2 + d^2])*ArcTanh[E^ (I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]]))/d + ((I + (c - d)/Sqrt[-c^ 2 + d^2])*(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*ArcTan[(d + (I*c + Sqrt[-c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + E^((2*I)*(e + f*x))])] - (I*c + Sqrt[-c^2 + d^2])*ArcTan h[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]]))/d)*Sqrt[g*Sin[e + f*x] ]*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[2]*d*E^(((5*I)/2)*(e + f*x))*(((-I)*(- 1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(5/2)*f*(Cos[(e + f*x)/2] + Sin [(e + f*x)/2])*Sqrt[Sin[e + f*x]])
Time = 0.85 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3042, 3407, 3042, 3254, 218, 3409, 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.
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c+d \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c+d \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3407 |
| \(\displaystyle \frac {g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)}}dx}{d}-\frac {c g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))}dx}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)}}dx}{d}-\frac {c g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))}dx}{d}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle -\frac {2 a g \int \frac {1}{\frac {\cos (e+f x) \cot (e+f x) a^2}{\sin (e+f x) a+a}+a}d\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {\sin (e+f x) a+a}}}{d f}-\frac {c g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))}dx}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {c g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c+d \sin (e+f x))}dx}{d}-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f}\) |
| \(\Big \downarrow \) 3409 |
| \(\displaystyle \frac {2 a c g \int \frac {1}{\frac {c \cos (e+f x) \cot (e+f x) a^2}{\sin (e+f x) a+a}+(c+d) a}d\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {\sin (e+f x) a+a}}}{d f}-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {c} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f \sqrt {c+d}}-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f}\) |
(-2*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(d*f) + (2*Sqrt[a]*Sqrt[c]*Sqrt[g]*ArcTa n[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]] *Sqrt[a + a*Sin[e + f*x]])])/(d*Sqrt[c + d]*f)
3.1.25.3.1 Defintions of rubi rules used
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]
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]
Int[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]])/((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g/d I nt[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x], x] - Simp[c*(g/d) I nt[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] , x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_. )*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-2*(b/f ) Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(934\) vs. \(2(117)=234\).
Time = 3.64 (sec) , antiderivative size = 935, normalized size of antiderivative = 6.28
1/2/f*(g*sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(2^(1/2)*(-(c-d)*(c+d) )^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1 /2)*ln(-(csc(f*x+e)-cot(f*x+e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)/(( csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1))+4*2^(1/2)*( -(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^(1 /2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)+4*2^(1/2)* (-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^( 1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-1)+2^(1/2)*( -(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^(1 /2)-d)*c)^(1/2)*ln(-((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot( f*x+e)-1)/(csc(f*x+e)-cot(f*x+e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)) +4*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*arctanh((csc(f* x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2))*c-4*(-(c-d)*( c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+ e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c+4*(((-(c-d)*(c+d))^(1/2) +d)*c)^(1/2)*arctanh((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2 )-d)*c)^(1/2))*c^2-4*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*arctanh((csc(f*x+e )-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2))*c*d+4*(((-(c-d)* (c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)* (c+d))^(1/2)+d)*c)^(1/2))*c^2-4*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arct...
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (117) = 234\).
Time = 1.47 (sec) , antiderivative size = 3273, normalized size of antiderivative = 21.97 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\text {Too large to display} \]
[1/4*(sqrt(-a*c*g/(c + d))*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a*c^4 + 192*a*c^3*d + 64*a*c^ 2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a *c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16 *c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 + 1 10*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4 + 149*c^3*d + 110*c^2*d^2 + 29*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e) - (51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c ^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^3 - (40*c^4 + 92*c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 41*c^2* d^2 + 11*c*d^3 + d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*c*g/(c + d))*sqr t(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2* d^2 + 4*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 3 2*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64*a*c^4 + 112*a*c^3*d + 56*a*c^2 *d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 - 2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2* d^2 + 18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d + 5 6*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*...
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {g \sin {\left (e + f x \right )}}}{c + d \sin {\left (e + f x \right )}}\, dx \]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{d \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c+d\,\sin \left (e+f\,x\right )} \,d x \]