Integrand size = 35, antiderivative size = 140 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}+\frac {\sqrt {2} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f} \]
-2*arctanh(cos(f*x+e)*a^(1/2)*c^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+ e))^(1/2))*c^(1/2)/f/a^(1/2)+arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^ (1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*2^(1/2)*(c-d)^(1/2)/f /a^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(665\) vs. \(2(140)=280\).
Time = 10.59 (sec) , antiderivative size = 665, normalized size of antiderivative = 4.75 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\csc (e+f x) \left (\sqrt {c} \log \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {2} \sqrt {c-d} \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {c} \log \left (d+\sqrt {2} \sqrt {c} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+c \tan \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {c} \log \left (c+\sqrt {2} \sqrt {c} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+d \tan \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {2} \sqrt {c-d} \log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a (1+\sin (e+f x))} \left (\sqrt {c} \csc (e+f x)+\frac {c \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )}}{2 \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {c-d} \sec ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {2} \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sqrt {c} \left (d \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {2} \sqrt {c} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}\right )}{2 \left (c+\sqrt {2} \sqrt {c} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+d \tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {\sqrt {2} \sqrt {c-d} \left (-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}\right )}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}\right )} \]
(Csc[e + f*x]*(Sqrt[c]*Log[Tan[(e + f*x)/2]] - Sqrt[2]*Sqrt[c - d]*Log[1 + Tan[(e + f*x)/2]] + Sqrt[c]*Log[d + Sqrt[2]*Sqrt[c]*Sqrt[(1 + Cos[e + f*x ])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + c*Tan[(e + f*x)/2]] - Sqrt[c]*Log[c + Sqrt[2]*Sqrt[c]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + d *Tan[(e + f*x)/2]] + Sqrt[2]*Sqrt[c - d]*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2 ]])*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a*(1 + Sin[e + f*x])]*(Sqrt[c]*Csc[e + f*x] + (c*Sqrt[Sec[(e + f*x)/2]^2])/(2*Sqrt[c + d*Sin[e + f*x]]) - (Sqr t[c - d]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[c]*( d*Sec[(e + f*x)/2]^2 + (Sqrt[2]*Sqrt[c]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]]))/(2*(c + Sq rt[2]*Sqrt[c]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + d*T an[(e + f*x)/2])) + (Sqrt[2]*Sqrt[c - d]*(-1/2*((c - d)*Sec[(e + f*x)/2]^2 ) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*S in[e + f*x]))/Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])) )
Time = 0.92 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3423, 3042, 3261, 221, 3422, 219}
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.
| \(\displaystyle \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a \sin (e+f x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{\sin (e+f x) \sqrt {a \sin (e+f x)+a}}dx\) |
| \(\Big \downarrow \) 3423 |
| \(\displaystyle \frac {c \int \frac {\csc (e+f x) \sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx}{a}-(c-d) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx}{a}-(c-d) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {2 a (c-d) \int \frac {1}{2 a^2-\frac {a^3 (c-d) \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}+\frac {c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx}{a}+\frac {\sqrt {2} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}\) |
| \(\Big \downarrow \) 3422 |
| \(\displaystyle \frac {\sqrt {2} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 c \int \frac {1}{1-\frac {a c \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {\cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {2} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}\) |
(-2*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) + (Sqrt[2]*Sqrt[c - d]*ArcTanh[ (Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[ c + d*Sin[e + f*x]])])/(Sqrt[a]*f)
3.1.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[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[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqr t[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subs t[Int[1/(1 - a*c*x^2), x], x, 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[b*c + a*d, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqr t[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(b*c - a*d)/c Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] + Simp[a /c Int[Sqrt[c + d*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^ 2, 0] && EqQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs. \(2(113)=226\).
Time = 1.80 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(\frac {\sqrt {c +d \sin \left (f x +e \right )}\, \sqrt {2}\, \left (\sqrt {2 c -2 d}\, \ln \left (\frac {2 \sqrt {2 c -2 d}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 \cos \left (f x +e \right ) d -2 c +2 d}{-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )}\right ) \sqrt {c}+\ln \left (\frac {\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+c \csc \left (f x +e \right )-c \cot \left (f x +e \right )+d}{\sqrt {c}}\right ) c -c \ln \left (-\frac {2 \left (\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )+c \sin \left (f x +e \right )-\cos \left (f x +e \right ) d +d \right )}{\cos \left (f x +e \right )-1}\right )\right ) \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right )}{2 f \left (1+\cos \left (f x +e \right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {c}}\) | \(326\) |
1/2/f*(c+d*sin(f*x+e))^(1/2)*2^(1/2)*((2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2 )*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)- d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c^ (1/2)+ln((c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)+c*csc(f* x+e)-c*cot(f*x+e)+d)/c^(1/2))*c-c*ln(-2*(c^(1/2)*2^(1/2)*((c+d*sin(f*x+e)) /(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-cos(f*x+e)*d+d)/(cos(f*x+e) -1)))*(cos(f*x+e)+sin(f*x+e)+1)/(1+cos(f*x+e))/(a*(1+sin(f*x+e)))^(1/2)/(( c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)/c^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (113) = 226\).
Time = 0.61 (sec) , antiderivative size = 2791, normalized size of antiderivative = 19.94 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Too large to display} \]
[1/4*(sqrt(2)*sqrt((c - d)/a)*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 + 4*sqrt(2)*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d )*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a) - (13*c^2 - 22*c*d - 3*d^2)* cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(9*c^2 - 14*c*d + 9*d^2)*cos(f* x + e) + ((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 + 2*(7*c^2 - 18*c*d + 7*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2* cos(f*x + e) - 4)) + sqrt(c/a)*log(((c^4 - 28*c^3*d + 70*c^2*d^2 - 28*c*d^ 3 + d^4)*cos(f*x + e)^5 - (31*c^4 - 196*c^3*d + 154*c^2*d^2 - 4*c*d^3 - d^ 4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(81*c^4 - 252*c^3*d + 150*c^2*d^2 - 28*c*d^3 + d^4)*cos(f*x + e)^3 + 2*(79*c^4 - 1 00*c^3*d + 74*c^2*d^2 - 4*c*d^3 - d^4)*cos(f*x + e)^2 - 8*((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e)^4 - 2*(5*c^3 - 14*c^2*d + 5*c*d^2)*cos(f*x + e)^3 + 51*c^3 - 59*c^2*d + 17*c*d^2 - d^3 - 2*(18*c^3 - 33*c^2*d + 12*c*d ^2 - d^3)*cos(f*x + e)^2 + 2*(13*c^3 - 14*c^2*d + 5*c*d^2)*cos(f*x + e) + ((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e)^3 - 51*c^3 + 59*c^2*d - 17*c *d^2 + d^3 + (11*c^3 - 35*c^2*d + 17*c*d^2 - d^3)*cos(f*x + e)^2 - (25*c^3 - 31*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(c/a) + (289*c^4 - 476*c^3*d + 230...
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sin {\left (e + f x \right )}}\, dx \]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )} \,d x } \]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]