Integrand size = 22, antiderivative size = 44 \[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \] Output:
(-6*cos(e*x+d)+8*sin(e*x+d))/e/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.70 \[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{e \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )} \] Input:
Integrate[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]
Output:
(-2*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])*Sqrt[5 + 4*Cos[d + e*x] + 3*Si n[d + e*x]])/(e*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2]))
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3042, 3591}
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.
\(\displaystyle \int \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}dx\) |
\(\Big \downarrow \) 3591 |
\(\displaystyle -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}\) |
Input:
Int[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]
Output:
(-2*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[ d + e*x]])
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] && EqQ[a^ 2 - b^2 - c^2, 0]
Time = 0.36 (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 | \(-\frac {5 i \sqrt {2}\, \sqrt {10+8 \cos \left (e x +d \right )+6 \sin \left (e x +d \right )}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}+30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}+40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}\, \left (5 \,{\mathrm e}^{i \left (e x +d \right )}-4-3 i\right ) \left (5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right )}{\left (25 \,{\mathrm e}^{2 i \left (e x +d \right )}+30 i {\mathrm e}^{i \left (e x +d \right )}+7+24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}+30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}+40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}}\) | \(228\) |
Input:
int((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x,method=_RETURNVERBOSE)
Output:
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
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.39 \[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}{3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e} \] Input:
integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="fricas")
Output:
-2*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5)*(cos(e*x + d) - 3*sin(e*x + d ) + 1)/(3*e*cos(e*x + d) + e*sin(e*x + d) + 3*e)
\[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int \sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5}\, dx \] Input:
integrate((5+4*cos(e*x+d)+3*sin(e*x+d))**(1/2),x)
Output:
Integral(sqrt(3*sin(d + e*x) + 4*cos(d + e*x) + 5), x)
\[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int { \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} \,d x } \] Input:
integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)
\[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int { \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} \,d x } \] Input:
integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)
Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2\,\sqrt {5}\,\left (3\,\cos \left (d+e\,x\right )-4\,\sin \left (d+e\,x\right )\right )}{5\,e\,\sqrt {\cos \left (d-\mathrm {atan}\left (\frac {3}{4}\right )+e\,x\right )+1}} \] Input:
int((4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(1/2),x)
Output:
-(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))
\[ \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\frac {-6 \sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}+20 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e +25 \left (\int \frac {\sqrt {4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}\, \cos \left (e x +d \right )}{4 \cos \left (e x +d \right )+3 \sin \left (e x +d \right )+5}d x \right ) e}{4 e} \] Input:
int((5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x)
Output:
( - 6*sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5) + 20*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)*e + 25*i nt((sqrt(4*cos(d + e*x) + 3*sin(d + e*x) + 5)*cos(d + e*x))/(4*cos(d + e*x ) + 3*sin(d + e*x) + 5),x)*e)/(4*e)