\(\int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx\) [420]

Optimal result

Integrand size = 22, antiderivative size = 48 \[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\frac {\sqrt {\frac {2}{5}} \text {arctanh}\left (\frac {\sin \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {1+\cos \left (d+e x-\arctan \left (\frac {3}{4}\right )\right )}}\right )}{e} \]

[Out]

1/5*arctanh(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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3194, 2728, 212} \[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\frac {\sqrt {\frac {2}{5}} \text {arctanh}\left (\frac {\sin \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )}{\sqrt {2} \sqrt {\cos \left (-\arctan \left (\frac {3}{4}\right )+d+e x\right )+1}}\right )}{e} \]

[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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

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 3194

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=-\frac {\left (\frac {2}{5}+\frac {6 i}{5}\right ) \sqrt {\frac {4}{5}+\frac {3 i}{5}} \arctan \left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {\frac {4}{5}+\frac {3 i}{5}} \left (-1+3 \tan \left (\frac {1}{4} (d+e x)\right )\right )\right ) \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )}{e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \]

[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]])

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {\left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5}\, \sqrt {10}\, \operatorname {arctanh}\left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5}\, \sqrt {10}}{10}\right )}{5 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(77\)
risch	\(\frac {2 i \left (5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right ) \sqrt {2}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+30 i {\mathrm e}^{i \left (e x +d \right )}+7+24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, {\mathrm e}^{-i \left (e x +d \right )}}{e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+30 i {\mathrm e}^{i \left (e x +d \right )}+7+24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {-\left (3 i {\mathrm e}^{2 i \left (e x +d \right )}-4 \,{\mathrm e}^{2 i \left (e x +d \right )}-4-3 i-10 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{-i \left (e x +d \right )}}}-\frac {2 i \left (5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right ) \left (\sqrt {5}\, \arctan \left (\frac {\sqrt {\left (2500-1875 i\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {5}}{125}\right ) \sqrt {\left (2500-1875 i\right ) {\mathrm e}^{i \left (e x +d \right )}}+125\right ) \sqrt {2}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+30 i {\mathrm e}^{i \left (e x +d \right )}+7+24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, {\mathrm e}^{-i \left (e x +d \right )}}{125 e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+30 i {\mathrm e}^{i \left (e x +d \right )}+7+24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{i \left (e x +d \right )}}\, \sqrt {-\left (3 i {\mathrm e}^{2 i \left (e x +d \right )}-4 \,{\mathrm e}^{2 i \left (e x +d \right )}-4-3 i-10 \,{\mathrm e}^{i \left (e x +d \right )}\right ) {\mathrm e}^{-i \left (e x +d \right )}}}\)	\(448\)

[In]

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

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (38) = 76\).

Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.06 \[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\frac {\sqrt {5} \sqrt {2} \log \left (-\frac {9 \, \cos \left (e x + d\right )^{2} + {\left (13 \, \cos \left (e x + d\right ) - 6\right )} \sin \left (e x + d\right ) + 2 \, {\left (\sqrt {5} \sqrt {2} \cos \left (e x + d\right ) - 3 \, \sqrt {5} \sqrt {2} \sin \left (e x + d\right ) + \sqrt {5} \sqrt {2}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} - 33 \, \cos \left (e x + d\right ) - 42}{9 \, \cos \left (e x + d\right )^{2} + {\left (13 \, \cos \left (e x + d\right ) + 14\right )} \sin \left (e x + d\right ) + 27 \, \cos \left (e x + d\right ) + 18}\right )}{10 \, e} \]

[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

Sympy [F]

\[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int { \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int { \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5}} \,d x \]

[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)