Integrand size = 29, antiderivative size = 73 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c-d} f} \]
-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)/f/a^(1/2)/(c-d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(73)=146\).
Time = 2.00 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.88 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \left (\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}\right )} \]
(Log[1 + Tan[(e + f*x)/2]] - 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]])/(f*Sqrt[ 3 + 3*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*(Sec[(e + f*x)/2]^2/(2 + 2*Ta n[(e + f*x)/2]) - (-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*Sin[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 = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3261, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3261 |
\(\displaystyle -\frac {2 a \int \frac {1}{2 a^2-\frac {a^3 (c-d) \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}}\) |
-((Sqrt[2]*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]*Sqrt[c - d]*f))
3.6.91.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(190\) vs. \(2(64)=128\).
Time = 9.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.62
method | result | size |
default | \(-\frac {\left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right ) \sqrt {c +d \sin \left (f x +e \right )}\, \ln \left (\frac {2 \sqrt {2 c -2 d}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )}\right ) \sqrt {2}}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2 c -2 d}}\) | \(191\) |
-1/f*(1+cos(f*x+e)+sin(f*x+e))*(c+d*sin(f*x+e))^(1/2)*ln(2*((2*c-2*d)^(1/2 )*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)- d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*2^ (1/2)/(cos(f*x+e)+1)/(a*(sin(f*x+e)+1))^(1/2)/((c+d*sin(f*x+e))/(cos(f*x+e )+1))^(1/2)/(2*c-2*d)^(1/2)
Time = 0.41 (sec) , antiderivative size = 475, normalized size of antiderivative = 6.51 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\left [\frac {\sqrt {2} \log \left (\frac {{\left (c^{2} - 14 \, c d + 17 \, d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, c^{2} - 22 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - \frac {4 \, \sqrt {2} {\left ({\left (c^{2} - 4 \, c d + 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} + 8 \, c d - 4 \, d^{2} - {\left (3 \, c^{2} - 4 \, c d + d^{2}\right )} \cos \left (f x + e\right ) + {\left (4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left (c^{2} - 4 \, c d + 3 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {a c - a d}} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (9 \, c^{2} - 14 \, c d + 9 \, d^{2}\right )} \cos \left (f x + e\right ) + {\left ({\left (c^{2} - 14 \, c d + 17 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} + 2 \, {\left (7 \, c^{2} - 18 \, c d + 7 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4}\right )}{4 \, \sqrt {a c - a d} f}, \frac {\sqrt {2} \sqrt {-\frac {1}{a c - a d}} \arctan \left (-\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left ({\left (c - 3 \, d\right )} \sin \left (f x + e\right ) - 3 \, c + d\right )} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {1}{a c - a d}}}{4 \, {\left (d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \]
[1/4*sqrt(2)*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 - (13*c^2 - 22*c* d - 3*d^2)*cos(f*x + e)^2 - 4*sqrt(2)*((c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^ 2 - 4*c^2 + 8*c*d - 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e) + (4*c^2 - 8*c*d + 4*d^2 + (c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*s in(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) - 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(a*c - a*d)*f), 1/2*sqrt(2)*sqrt(-1/(a*c - a*d))*arctan(-1/4*sqrt(2)*sqrt(a*sin(f *x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)*s qrt(-1/(a*c - a*d))/(d*cos(f*x + e)*sin(f*x + e) + c*cos(f*x + e)))/f]
\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]