Integrand size = 33, antiderivative size = 433 \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=-\frac {b \sqrt {g} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}+\frac {b^{3/2} \sqrt {g} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \sqrt [4]{-a^2+b^2} f}+\frac {b \sqrt {g} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}-\frac {b^{3/2} \sqrt {g} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \sqrt [4]{-a^2+b^2} f}-\frac {(g \cos (e+f x))^{3/2} \csc (e+f x)}{a f g}-\frac {\sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {\cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}} \]
-(g*cos(f*x+e))^(3/2)*csc(f*x+e)/a/f/g-b*arctan((g*cos(f*x+e))^(1/2)/g^(1/ 2))*g^(1/2)/a^2/f+b^(3/2)*arctan(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^( 1/4)/g^(1/2))*g^(1/2)/a^2/(-a^2+b^2)^(1/4)/f+b*arctanh((g*cos(f*x+e))^(1/2 )/g^(1/2))*g^(1/2)/a^2/f-b^(3/2)*arctanh(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^ 2+b^2)^(1/4)/g^(1/2))*g^(1/2)/a^2/(-a^2+b^2)^(1/4)/f+b*g*(cos(1/2*f*x+1/2* e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(b-(-a^2+ b^2)^(1/2)),2^(1/2))*cos(f*x+e)^(1/2)/a/f/(b-(-a^2+b^2)^(1/2))/(g*cos(f*x+ e))^(1/2)+b*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(s in(1/2*f*x+1/2*e),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(f*x+e)^(1/2)/a/f/( b+(-a^2+b^2)^(1/2))/(g*cos(f*x+e))^(1/2)-(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos( 1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*(g*cos(f*x+e))^(1/2)/ a/f/cos(f*x+e)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 34.22 (sec) , antiderivative size = 1550, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx =\text {Too large to display} \]
-((Sqrt[g*Cos[e + f*x]]*Cot[e + f*x])/(a*f)) + (Sqrt[g*Cos[e + f*x]]*((4*a *(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f *x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^ 2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^ 2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)* Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sq rt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b] *(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) + ( 5*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6 *Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2 )*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f* x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b ]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4 )*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/ 4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e +...
Time = 1.12 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3042, 3377, 2009}
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.
| \(\displaystyle \int \frac {\csc ^2(e+f x) \sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{\sin (e+f x)^2 (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3377 |
| \(\displaystyle \int \left (\frac {b^2 \sqrt {g \cos (e+f x)}}{a^2 (a+b \sin (e+f x))}-\frac {b \csc (e+f x) \sqrt {g \cos (e+f x)}}{a^2}+\frac {\csc ^2(e+f x) \sqrt {g \cos (e+f x)}}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^{3/2} \sqrt {g} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f \sqrt [4]{b^2-a^2}}-\frac {b \sqrt {g} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}-\frac {b^{3/2} \sqrt {g} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f \sqrt [4]{b^2-a^2}}+\frac {b \sqrt {g} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x) (g \cos (e+f x))^{3/2}}{a f g}-\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f \sqrt {\cos (e+f x)}}\) |
-((b*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + (b^(3/2)*Sqr t[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/ (a^2*(-a^2 + b^2)^(1/4)*f) + (b*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[ g]])/(a^2*f) - (b^(3/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((- a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) - ((g*Cos[e + f*x]) ^(3/2)*Csc[e + f*x])/(a*f*g) - (Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2 , 2])/(a*f*Sqrt[Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/ (b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b - Sqrt[-a^2 + b^2])*f*Sqrt[ g*Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x] ])
3.14.75.3.1 Defintions of rubi rules used
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a _) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/ 2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.10 (sec) , antiderivative size = 1266, normalized size of antiderivative = 2.92
(4*b*g*(1/4/a^2/(-g)^(1/2)*ln((-2*g+2*(-g)^(1/2)*(2*g*cos(1/2*f*x+1/2*e)^2 -g)^(1/2))/cos(1/2*f*x+1/2*e))+1/8/a^2/g^(1/2)*ln((-4*g*cos(1/2*f*x+1/2*e) +2*g^(1/2)*(-2*g*sin(1/2*f*x+1/2*e)^2+g)^(1/2)-2*g)/(cos(1/2*f*x+1/2*e)+1) )+1/8/a^2/g^(1/2)*ln((4*g*cos(1/2*f*x+1/2*e)+2*g^(1/2)*(-2*g*sin(1/2*f*x+1 /2*e)^2+g)^(1/2)-2*g)/(-1+cos(1/2*f*x+1/2*e)))+1/16/a^2/(g^2*(a^2-b^2)/b^2 )^(1/4)*2^(1/2)*(ln((2*g*cos(1/2*f*x+1/2*e)^2-g-(g^2*(a^2-b^2)/b^2)^(1/4)* (2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2))/(2*g *cos(1/2*f*x+1/2*e)^2-g+(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^ 2-g)^(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*g*cos( 1/2*f*x+1/2*e)^2-g)^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/4))/(g^2*(a^2-b^2)/b^2)^( 1/4))+2*arctan((2^(1/2)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)-(g^2*(a^2-b^2)/ b^2)^(1/4))/(g^2*(a^2-b^2)/b^2)^(1/4))))-1/8*(g*(2*cos(1/2*f*x+1/2*e)^2-1) *sin(1/2*f*x+1/2*e)^2)^(1/2)/a*g/cos(1/2*f*x+1/2*e)/(-2*g*sin(1/2*f*x+1/2* e)^4+g*sin(1/2*f*x+1/2*e)^2)^(3/2)*(-8*cos(1/2*f*x+1/2*e)*EllipticE(cos(1/ 2*f*x+1/2*e),2^(1/2))*(2*sin(1/2*f*x+1/2*e)^2-1)^(3/2)*(sin(1/2*f*x+1/2*e) ^2)^(1/2)*sin(1/2*f*x+1/2*e)^2*g-32*sin(1/2*f*x+1/2*e)^8*g+48*g*sin(1/2*f* x+1/2*e)^6+cos(1/2*f*x+1/2*e)*(-2*g*sin(1/2*f*x+1/2*e)^4+g*sin(1/2*f*x+1/2 *e)^2)^(3/2)/a^2*sum(1/_alpha*(8*(g*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)* (sin(1/2*f*x+1/2*e)^2)^(1/2)*(2*sin(1/2*f*x+1/2*e)^2-1)^(1/2)*EllipticPi(c os(1/2*f*x+1/2*e),(-4*_alpha^2*b^2+4*b^2)/a^2,2^(1/2))*_alpha^3*b^2-8*b...
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g \cos {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}}{a + b \sin {\left (e + f x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \csc \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}}{{\sin \left (e+f\,x\right )}^2\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]