3.427 \(\int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{e} \]

[Out]

-1/5*arctan(1/2*sin(d+e*x-arctan(3/4))*2^(1/2)/(-1+cos(d+e*x-arctan(3/4)))^(1/2))*10^(1/2)/e

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3115, 2649, 204} \[ -\frac {\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

-((Sqrt[2/5]*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/e)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3115

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx &=\int \frac {1}{\sqrt {-5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}} \, dx\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-10-x^2} \, dx,x,-\frac {5 \sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {-5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{e}\\ &=-\frac {\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{e}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 99, normalized size = 2.02 \[ \frac {\left (\frac {2}{5}+\frac {6 i}{5}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \tanh ^{-1}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (\tan \left (\frac {1}{4} (d+e x)\right )+3\right )\right )}{e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

((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]])

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fricas [B]  time = 2.11, size = 88, normalized size = 1.80 \[ \frac {\sqrt {5} \sqrt {2} \arctan \left (-\frac {{\left (3 \, \sqrt {5} \sqrt {2} \cos \left (e x + d\right ) + \sqrt {5} \sqrt {2} \sin \left (e x + d\right ) + 3 \, \sqrt {5} \sqrt {2}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{10 \, {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}\right )}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="fricas")

[Out]

1/5*sqrt(5)*sqrt(2)*arctan(-1/10*(3*sqrt(5)*sqrt(2)*cos(e*x + d) + sqrt(5)*sqrt(2)*sin(e*x + d) + 3*sqrt(5)*sq
rt(2))*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5)/(cos(e*x + d) - 3*sin(e*x + d) + 1))/e

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5), x)

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maple [A]  time = 0.33, size = 77, normalized size = 1.57 \[ \frac {\left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )}{5 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x)

[Out]

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+arctan(4/3))/(-5+5*sin(e*x+d+arctan(4/3)))^(1/2)/e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(1/2),x)

[Out]

int(1/(4*cos(d + e*x) + 3*sin(d + e*x) - 5)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))**(1/2),x)

[Out]

Integral(1/sqrt(3*sin(d + e*x) + 4*cos(d + e*x) - 5), x)

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