Integrand size = 22, antiderivative size = 49 \[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=-\frac {\sqrt {\frac {2}{5}} \arctan \left (\frac {\sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{e} \] Output:
-1/5*10^(1/2)*arctan(1/2*sin(d+e*x-arctan(3/4))*2^(1/2)/(-1+cos(d+e*x-arct an(3/4)))^(1/2))/e
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\frac {\left (\frac {2}{5}+\frac {6 i}{5}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \text {arctanh}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (3+\tan \left (\frac {1}{4} (d+e x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )}{e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \] Input:
Integrate[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]
Output:
((2/5 + (6*I)/5)*Sqrt[-4/5 - (3*I)/5]*ArcTanh[(1/10 + (3*I)/10)*Sqrt[-4/5 - (3*I)/5]*(3 + Tan[(d + e*x)/4])]*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2]) )/(e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3042, 3594, 3042, 3128, 217}
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.
| \(\displaystyle \int \frac {1}{\sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}}dx\) |
| \(\Big \downarrow \) 3594 |
| \(\displaystyle \int \frac {1}{\sqrt {5 \cos \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )-5}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {5 \sin \left (-\arctan \left (\frac {3}{4}\right )+d+e x+\frac {\pi }{2}\right )-5}}dx\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {2 \int \frac {1}{-\frac {5 \sin ^2\left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )-1}-10}d\left (-\frac {\sqrt {5} \sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )-1}}\right )}{e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\sqrt {\frac {2}{5}} \arctan \left (\frac {\sin \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )}{\sqrt {2} \sqrt {\cos \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )-1}}\right )}{e}\) |
Input:
Int[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]
Output:
-((Sqrt[2/5]*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/e)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
Time = 0.94 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(\frac {\left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )}{5 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\) | \(77\) |
| risch | \(\frac {2 i \left (5 \,{\mathrm e}^{i \left (e x +d \right )}-4-3 i\right ) \sqrt {2}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+7+24 i-30 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, {\mathrm e}^{-i \left (e x +d \right )}}{e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+7+24 i-30 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {-\left (3 i {\mathrm e}^{2 i \left (e x +d \right )}-4 \,{\mathrm e}^{2 i \left (e x +d \right )}-4-3 i+10 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{-i \left (e x +d \right )}}}-\frac {2 i \left (-5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right ) \left (\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (2500-1875 i\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {5}}{125}\right ) \sqrt {\left (2500-1875 i\right ) {\mathrm e}^{i \left (e x +d \right )}}-125\right ) \sqrt {2}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+7+24 i-30 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, {\mathrm e}^{-i \left (e x +d \right )}}{125 e \sqrt {\left (-100+75 i\right ) \left (-25 \,{\mathrm e}^{2 i \left (e x +d \right )}-7-24 i+30 i {\mathrm e}^{i \left (e x +d \right )}+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {-\left (3 i {\mathrm e}^{2 i \left (e x +d \right )}-4 \,{\mathrm e}^{2 i \left (e x +d \right )}-4-3 i+10 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{-i \left (e x +d \right )}}}\) | \(448\) |
Input:
int(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*(sin(e*x+d+arctan(4/3))-1)*(-5*sin(e*x+d+arctan(4/3))-5)^(1/2)*10^(1/2 )*arctan(1/10*(-5*sin(e*x+d+arctan(4/3))-5)^(1/2)*10^(1/2))/cos(e*x+d+arct an(4/3))/(-5+5*sin(e*x+d+arctan(4/3)))^(1/2)/e
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\frac {\sqrt {\frac {2}{5}} \arctan \left (-\frac {5 \, \sqrt {\frac {2}{5}} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{3 \, \cos \left (e x + d\right ) - 4 \, \sin \left (e x + d\right )}\right )}{e} \] Input:
integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="fricas")
Output:
sqrt(2/5)*arctan(-5*sqrt(2/5)*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5)/(3 *cos(e*x + d) - 4*sin(e*x + d)))/e
\[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5}}\, dx \] Input:
integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))**(1/2),x)
Output:
Integral(1/sqrt(3*sin(d + e*x) + 4*cos(d + e*x) - 5), x)
\[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int { \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}} \,d x } \] Input:
integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5), x)
\[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int { \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}} \,d x } \] Input:
integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5), x)
Timed out. \[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5}} \,d x \] Input:
int(1/(4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(1/2),x)
Output:
int(1/(4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(1/2), x)
\[ \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )-5}d x \] Input:
int(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x)
Output:
int(sqrt(4*cos(d + e*x) + 3*sin(d + e*x) - 5)/(4*cos(d + e*x) + 3*sin(d + e*x) - 5),x)