\(\int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx\) [426]

Optimal result

Integrand size = 22, antiderivative size = 44 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3191} \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}} \]

[In]

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

[Out]

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

Rule 3191

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[-2*((c*Cos[d
+ e*x] - b*Sin[d + e*x])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.70 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\frac {2 \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right ) \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}}{e \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {10 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right )}{\cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(50\)
risch	\(\frac {5 i \sqrt {2}\, \sqrt {-10+8 \cos \left (e x +d \right )+6 \sin \left (e x +d \right )}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}-30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}\, \left (5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right ) \left (5 \,{\mathrm e}^{i \left (e x +d \right )}-4-3 i\right )}{\left (30 i {\mathrm e}^{i \left (e x +d \right )}-25 \,{\mathrm e}^{2 i \left (e x +d \right )}-7-24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}-30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}}\)	\(228\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\frac {2 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} {\left (3 \, \cos \left (e x + d\right ) + \sin \left (e x + d\right ) + 3\right )}}{e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + e} \]

[In]

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

[Out]

2*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) - 5)*(3*cos(e*x + d) + sin(e*x + d) + 3)/(e*cos(e*x + d) - 3*e*sin(e*x
+ d) + e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2\,\sqrt {5}\,\left (3\,\cos \left (d+e\,x\right )-4\,\sin \left (d+e\,x\right )\right )}{5\,e\,\sqrt {\cos \left (d-\mathrm {atan}\left (\frac {3}{4}\right )+e\,x\right )-1}} \]

[In]

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

[Out]

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