\(\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx\) [418]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e} \]

[Out]

-40/3*(3*cos(e*x+d)-4*sin(e*x+d))/e/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)-2/3*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*c
os(e*x+d)+3*sin(e*x+d))^(1/2)/e

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 93, 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 = {3192, 3191} \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=-\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

[In]

Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]

[Out]

(-40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] -
4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e)

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]

Rule 3192

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(c*Cos[d
+ e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Dist[a*((2*n - 1)/n), In
t[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0
] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}+\frac {20}{3} \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx \\ & = -\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.12 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\frac {(5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \left (-45 \cos \left (\frac {1}{2} (d+e x)\right )-13 \cos \left (\frac {3}{2} (d+e x)\right )+9 \left (15 \sin \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {3}{2} (d+e x)\right )\right )\right )}{3 e \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^3} \]

[In]

Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]

[Out]

((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2)*(-45*Cos[(d + e*x)/2] - 13*Cos[(3*(d + e*x))/2] + 9*(15*Sin[(d +
e*x)/2] + Sin[(3*(d + e*x))/2])))/(3*e*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3)

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {50 \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(60\)

[In]

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

[Out]

50/3*(1+sin(e*x+d+arctan(4/3)))*(sin(e*x+d+arctan(4/3))-1)*(sin(e*x+d+arctan(4/3))+5)/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 = 81, normalized size of antiderivative = 0.87 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=-\frac {2 \, {\left (13 \, \cos \left (e x + d\right )^{2} - 9 \, {\left (\cos \left (e x + d\right ) + 8\right )} \sin \left (e x + d\right ) + 29 \, \cos \left (e x + d\right ) + 16\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{3 \, {\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \]

[In]

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

[Out]

-2/3*(13*cos(e*x + d)^2 - 9*(cos(e*x + d) + 8)*sin(e*x + d) + 29*cos(e*x + d) + 16)*sqrt(4*cos(e*x + d) + 3*si
n(e*x + d) + 5)/(3*e*cos(e*x + d) + e*sin(e*x + d) + 3*e)

Sympy [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\int \left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((3*sin(d + e*x) + 4*cos(d + e*x) + 5)**(3/2), x)

Maxima [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx=\int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5\right )}^{3/2} \,d x \]

[In]

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

[Out]

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