\(\int \sqrt {3+4 \cos (c+d x)} \, dx\) [512]

Optimal result

Integrand size = 14, antiderivative size = 23 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]

[Out]

2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2732} \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]

[In]

Int[Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(2*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \]

[In]

Integrate[Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(2*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(136\) vs. \(2(47)=94\).

Time = 3.66 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.96

method	result	size
default	\(\frac {2 \sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(137\)
risch	\(-\frac {2 i \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d}-\frac {i \left (\frac {6 \left (\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}\, \sqrt {14}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{-\frac {3}{4}-\frac {i \sqrt {7}}{4}}}\, F\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )}{7 \sqrt {2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}}}-\frac {4 \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}{\sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {8 \left (\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}\, \sqrt {14}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{-\frac {3}{4}-\frac {i \sqrt {7}}{4}}}\, \left (-\frac {i \sqrt {7}\, E\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )}{2}+\left (-\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) F\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{4}+\frac {i \sqrt {7}}{4}}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )\right )}{7 \sqrt {2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}\)	\(659\)

[In]

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

[Out]

2*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-8*cos(1/2*d*x+1/2*c)
^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2))/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1
/2*d*x+1/2*c)/(8*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 4 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 4 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{2 \, d} \]

[In]

integrate((3+4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/2*(-I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) + I*sqrt(2)*weierstrassPInvers
e(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2) + 4*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1,
cos(d*x + c) + I*sin(d*x + c) + 1/2)) - 4*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x
+ c) - I*sin(d*x + c) + 1/2)))/d

Sympy [F]

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

[In]

integrate((3+4*cos(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(4*cos(c + d*x) + 3), x)

Maxima [F]

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

[In]

integrate((3+4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*cos(d*x + c) + 3), x)

Giac [F]

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

[In]

integrate((3+4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*cos(d*x + c) + 3), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\int \sqrt {4\,\cos \left (c+d\,x\right )+3} \,d x \]

[In]

int((4*cos(c + d*x) + 3)^(1/2),x)

[Out]

int((4*cos(c + d*x) + 3)^(1/2), x)