Integrand size = 40, antiderivative size = 103 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c f} \]
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)/c/f+2*sec(f*x+e)*(g*sin(f*x+e))^(1/2)*(a+a*sin(f*x+ e))^(1/2)/c/f
Time = 2.59 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sec (e+f x) \left (\arcsin \left (\sqrt {1-\sin (e+f x)}\right ) \sqrt {1-\sin (e+f x)}+\sqrt {\sin (e+f x)}\right ) \sqrt {g \sin (e+f x)} \sqrt {a (1+\sin (e+f x))}}{c f \sqrt {\sin (e+f x)}} \]
(2*Sec[e + f*x]*(ArcSin[Sqrt[1 - Sin[e + f*x]]]*Sqrt[1 - Sin[e + f*x]] + S qrt[Sin[e + f*x]])*Sqrt[g*Sin[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])])/(c*f*S qrt[Sin[e + f*x]])
Time = 0.85 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3042, 3407, 3042, 3254, 218, 3409, 15}
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.
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c-c \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c-c \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3407 |
| \(\displaystyle g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))}dx-\frac {g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)}}dx}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))}dx-\frac {g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)}}dx}{c}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {2 a g \int \frac {1}{\frac {\cos (e+f x) \cot (e+f x) a^2}{\sin (e+f x) a+a}+a}d\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {\sin (e+f x) a+a}}}{c f}+g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle g \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))}dx+\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{c f}\) |
| \(\Big \downarrow \) 3409 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{c f}-\frac {2 a g \int \frac {\sec (e+f x) (\sin (e+f x) a+a) \tan (e+f x)}{a^2 c}d\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {\sin (e+f x) a+a}}}{f}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c f}\) |
(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]])])/(c*f) + (2*Sec[e + f*x]*Sqrt[g*Sin[e + f* x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f)
3.1.15.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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. \(753\) vs. \(2(87)=174\).
Time = 3.21 (sec) , antiderivative size = 754, normalized size of antiderivative = 7.32
| method | result | size |
| default | \(-\frac {\sqrt {\frac {g \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1\right ) \sqrt {\frac {a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+2 \csc \left (f x +e \right )-2 \cot \left (f x +e \right )+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}}\, \left (\sqrt {2}\, \ln \left (-\frac {\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-1\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+\sqrt {2}\, \ln \left (-\frac {\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}{\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}\right ) \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )-\sqrt {2}\, \ln \left (-\frac {\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}\right )-4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1\right )-4 \sqrt {2}\, \arctan \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-1\right )-\sqrt {2}\, \ln \left (-\frac {\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}-\csc \left (f x +e \right )+\cot \left (f x +e \right )-1}{\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \sqrt {2}+1}\right )+8 \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\right ) \sqrt {2}}{4 c f \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}\, \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}+1\right ) \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )}-1\right )}\) | \(754\) |
-1/4/c/f*(g/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(1/ 2)*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(a*((1-cos(f*x+e))^2*csc(f*x+e)^2+2*c sc(f*x+e)-2*cot(f*x+e)+1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(1/2)/(-cot(f *x+e)+csc(f*x+e)+1)*(2^(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)+c ot(f*x+e)-1))*(csc(f*x+e)-cot(f*x+e))+4*2^(1/2)*arctan((csc(f*x+e)-cot(f*x +e))^(1/2)*2^(1/2)+1)*(csc(f*x+e)-cot(f*x+e))+4*2^(1/2)*arctan((csc(f*x+e) -cot(f*x+e))^(1/2)*2^(1/2)-1)*(csc(f*x+e)-cot(f*x+e))+2^(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))*(csc(f*x+e)-cot(f*x+e))-2^( 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)* arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)-4*2^(1/2)*arctan((csc(f*x+ e)-cot(f*x+e))^(1/2)*2^(1/2)-1)-2^(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))+8*(csc(f*x+e)-cot(f*x+e))^(1/2))/(csc(f*x+e)-cot( f*x+e))^(1/2)/((csc(f*x+e)-cot(f*x+e))^(1/2)+1)/((csc(f*x+e)-cot(f*x+e))^( 1/2)-1)*2^(1/2)
Time = 0.41 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.29 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\left [\frac {\sqrt {-a g} \cos \left (f x + e\right ) \log \left (\frac {128 \, a g \cos \left (f x + e\right )^{5} - 128 \, a g \cos \left (f x + e\right )^{4} - 416 \, a g \cos \left (f x + e\right )^{3} + 128 \, a g \cos \left (f x + e\right )^{2} + 289 \, a g \cos \left (f x + e\right ) + 8 \, {\left (16 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} - 66 \, \cos \left (f x + e\right )^{2} + {\left (16 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 51\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 51\right )} \sqrt {-a g} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )} + a g + {\left (128 \, a g \cos \left (f x + e\right )^{4} + 256 \, a g \cos \left (f x + e\right )^{3} - 160 \, a g \cos \left (f x + e\right )^{2} - 288 \, a g \cos \left (f x + e\right ) + a g\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{4 \, c f \cos \left (f x + e\right )}, -\frac {\sqrt {a g} \arctan \left (\frac {\sqrt {a g} {\left (8 \, \cos \left (f x + e\right )^{2} + 8 \, \sin \left (f x + e\right ) - 9\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{4 \, {\left (2 \, a g \cos \left (f x + e\right )^{3} + a g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a g \cos \left (f x + e\right )\right )}}\right ) \cos \left (f x + e\right ) - 4 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{2 \, c f \cos \left (f x + e\right )}\right ] \]
[1/4*(sqrt(-a*g)*cos(f*x + e)*log((128*a*g*cos(f*x + e)^5 - 128*a*g*cos(f* x + e)^4 - 416*a*g*cos(f*x + e)^3 + 128*a*g*cos(f*x + e)^2 + 289*a*g*cos(f *x + e) + 8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 + ( 16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(-a*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f* x + e)) + a*g + (128*a*g*cos(f*x + e)^4 + 256*a*g*cos(f*x + e)^3 - 160*a*g *cos(f*x + e)^2 - 288*a*g*cos(f*x + e) + a*g)*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(c *f*cos(f*x + e)), -1/2*(sqrt(a*g)*arctan(1/4*sqrt(a*g)*(8*cos(f*x + e)^2 + 8*sin(f*x + e) - 9)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(2*a*g* cos(f*x + e)^3 + a*g*cos(f*x + e)*sin(f*x + e) - 2*a*g*cos(f*x + e)))*cos( f*x + e) - 4*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(c*f*cos(f*x + e))]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=- \frac {\int \frac {\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a \sin {\left (e + f x \right )} + a}}{\sin {\left (e + f x \right )} - 1}\, dx}{c} \]
\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int { -\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) - c} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c-c\,\sin \left (e+f\,x\right )} \,d x \]