Integrand size = 35, antiderivative size = 123 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {a} \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f}-\frac {2 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f} \]
-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))*a^(1/2)*c^(1/2)/f-2*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin( f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*a^(1/2)*d^(1/2)/f
Result contains complex when optimal does not.
Time = 2.90 (sec) , antiderivative size = 567, normalized size of antiderivative = 4.61 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {\left (\sqrt {c} \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {i e}{2}} \left (-\sqrt {2} c \left (-1+e^{i (e+f x)}\right )-i \sqrt {2} d \left (1+e^{i (e+f x)}\right )+2 i \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{c^{3/2} \left (1+e^{i (e+f x)}\right )}\right )+\sqrt {c} \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {i e}{2}} \left (-i \sqrt {2} d \left (-1+e^{i (e+f x)}\right )+\sqrt {2} c \left (1+e^{i (e+f x)}\right )+2 \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{c^{3/2} \left (-1+e^{i (e+f x)}\right )}\right )-i \sqrt {d} \left (\log \left (\frac {2 e^{-\frac {1}{2} i (e+2 f x)} \left ((-1)^{3/4} d+\sqrt [4]{-1} c e^{i (e+f x)}+i \sqrt {d} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{d^{3/2}}\right )-\log \left (\frac {(1+i) \sqrt {2} \left (c-i d \cos (e+f x)+d \sin (e+f x)+(1-i) \sqrt {d} \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}\right )}{\sqrt {d}}\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \sqrt {c+d \sin (e+f x)}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}} \]
-(((Sqrt[c]*Log[((1/2 + I/2)*(-(Sqrt[2]*c*(-1 + E^(I*(e + f*x)))) - I*Sqrt [2]*d*(1 + E^(I*(e + f*x))) + (2*I)*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d *(-1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(1 + E^(I*(e + f*x)) ))] + Sqrt[c]*Log[((1/2 + I/2)*((-I)*Sqrt[2]*d*(-1 + E^(I*(e + f*x))) + Sq rt[2]*c*(1 + E^(I*(e + f*x))) + 2*Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*( -1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(-1 + E^(I*(e + f*x))) )] - I*Sqrt[d]*(Log[(2*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sq rt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(d^(3 /2)*E^((I/2)*(e + 2*f*x)))] - Log[((1 + I)*Sqrt[2]*(c - I*d*Cos[e + f*x] + d*Sin[e + f*x] + (1 - I)*Sqrt[d]*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])]))/Sqrt[d]]))*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sq rt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x] )]))
Time = 0.88 (sec) , antiderivative size = 123, 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, 3428, 3042, 3254, 218, 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.
\(\displaystyle \int \csc (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}{\sin (e+f x)}dx\) |
\(\Big \downarrow \) 3428 |
\(\displaystyle d \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx+c \int \frac {\csc (e+f x) \sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx+c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3254 |
\(\displaystyle c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx-\frac {2 a d \int \frac {1}{\frac {a^2 d \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}+a}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle c \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x) \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \sqrt {a} \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}\) |
\(\Big \downarrow \) 3422 |
\(\displaystyle -\frac {2 a c \int \frac {1}{1-\frac {a c \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {\cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}-\frac {2 \sqrt {a} \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \sqrt {a} \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}-\frac {2 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}\) |
(-2*Sqrt[a]*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[ e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f - (2*Sqrt[a]*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 ]])])/f
3.1.35.3.1 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[((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[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[(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_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]])/sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[d Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[c Int[Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*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] || NeQ[c^2 - d^2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(5065\) vs. \(2(99)=198\).
Time = 0.84 (sec) , antiderivative size = 5066, normalized size of antiderivative = 41.19
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (99) = 198\).
Time = 1.02 (sec) , antiderivative size = 3539, normalized size of antiderivative = 28.77 \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\text {Too large to display} \]
[1/4*(sqrt(a*c)*log(((a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*c*d^3 + 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 - (31*a*c^4 - 196*a*c^3*d + 154*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e) ^4 - 2*(81*a*c^4 - 252*a*c^3*d + 150*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f *x + e)^3 + 2*(79*a*c^4 - 100*a*c^3*d + 74*a*c^2*d^2 - 4*a*c*d^3 - a*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*c)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) + (289*a*c^4 - 476*a*c^3*d + 230*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(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 + (a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^4 + 32*(a*c^4 - 7*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e)^3 - 2*(65*a*c^4 - 140*a *c^3*d + 38*a*c^2*d^2 - 12*a*c*d^3 + a*d^4)*cos(f*x + e)^2 - 32*(9*a*c^4 - 15*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^5 + cos(f*x + e)^4 - 2*cos(f*x + e)^3 - 2*cos(f*x + e)^2 + (cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sin(f*x + e) + cos(f*x + e) + 1)) + sqrt(...
\[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )}}\, dx \]
\[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{\sin \left (f x + e\right )} \,d x } \]
Timed out. \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )} \,d x \]