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

Optimal. Leaf size=48 \[ \frac{\sqrt{\frac{2}{5}} \tanh ^{-1}\left (\frac{\sin \left (d+e x-\tan ^{-1}\left (\frac{3}{4}\right )\right )}{\sqrt{2} \sqrt{\cos \left (d+e x-\tan ^{-1}\left (\frac{3}{4}\right )\right )+1}}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0648061, antiderivative size = 48, normalized size of antiderivative = 1., 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, 206} \[ \frac{\sqrt{\frac{2}{5}} \tanh ^{-1}\left (\frac{\sin \left (d+e x-\tan ^{-1}\left (\frac{3}{4}\right )\right )}{\sqrt{2} \sqrt{\cos \left (d+e x-\tan ^{-1}\left (\frac{3}{4}\right )\right )+1}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 206

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

Mathematica [C]  time = 0.105108, size = 101, normalized size = 2.1 \[ -\frac{\left (\frac{2}{5}+\frac{6 i}{5}\right ) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \tan ^{-1}\left (\left (\frac{1}{10}+\frac{3 i}{10}\right ) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \left (3 \tan \left (\frac{1}{4} (d+e x)\right )-1\right )\right ) \left (\sin \left (\frac{1}{2} (d+e x)\right )+3 \cos \left (\frac{1}{2} (d+e x)\right )\right )}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

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]*ArcTan[(1/10 + (3*I)/10)*Sqrt[4/5 + (3*I)/5]*(-1 + 3*Tan[(d + e*x)/4])]*
(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])

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Maple [A]  time = 0.89, size = 77, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 1+\sin \left ( ex+d+\arctan \left ({\frac{4}{3}} \right ) \right ) \right ) \sqrt{10}}{5\,\cos \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) e}\sqrt{-5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) +5}{\it Artanh} \left ({\frac{\sqrt{10}}{10}\sqrt{-5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) +5}} \right ){\frac{1}{\sqrt{5+5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) }}}} \end{align*}

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*(1+sin(e*x+d+arctan(4/3)))*(-5*sin(e*x+d+arctan(4/3))+5)^(1/2)*10^(1/2)*arctanh(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.77431, size = 428, normalized size = 8.92 \begin{align*} \frac{\sqrt{5} \sqrt{2} \log \left (-\frac{9 \, \cos \left (e x + d\right )^{2} +{\left (13 \, \cos \left (e x + d\right ) - 6\right )} \sin \left (e x + d\right ) + 2 \,{\left (\sqrt{5} \sqrt{2} \cos \left (e x + d\right ) - 3 \, \sqrt{5} \sqrt{2} \sin \left (e x + d\right ) + \sqrt{5} \sqrt{2}\right )} \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} - 33 \, \cos \left (e x + d\right ) - 42}{9 \, \cos \left (e x + d\right )^{2} +{\left (13 \, \cos \left (e x + d\right ) + 14\right )} \sin \left (e x + d\right ) + 27 \, \cos \left (e x + d\right ) + 18}\right )}{10 \, e} \end{align*}

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]

1/10*sqrt(5)*sqrt(2)*log(-(9*cos(e*x + d)^2 + (13*cos(e*x + d) - 6)*sin(e*x + d) + 2*(sqrt(5)*sqrt(2)*cos(e*x
+ d) - 3*sqrt(5)*sqrt(2)*sin(e*x + d) + sqrt(5)*sqrt(2))*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5) - 33*cos(e*
x + d) - 42)/(9*cos(e*x + d)^2 + (13*cos(e*x + d) + 14)*sin(e*x + d) + 27*cos(e*x + d) + 18))/e

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \sin{\left (d + e x \right )} + 4 \cos{\left (d + e x \right )} + 5}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}\,{d x} \end{align*}

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)