\(\int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) [446]

Optimal result

Integrand size = 33, antiderivative size = 74 \[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \]

[Out]

2/3*cot(d*x+c)*EllipticE((3-2*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),1/5*I*5^(1/2))*5^(1/2)*(1-sec(d*x+c))^(1/2)*(
1+sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {3073} \[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {5} \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{3 d} \]

[In]

Int[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]

[Out]

(2*Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c +
d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)

Rule 3073

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e +
f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e +
f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \\ \end{align*}

Mathematica [F]

\[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \]

[In]

Integrate[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]

[Out]

Integrate[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)), x]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (64 ) = 128\).

Time = 11.59 (sec) , antiderivative size = 485, normalized size of antiderivative = 6.55

method	result	size
parts	\(\frac {\left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )}{d \left (-3+2 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}-\frac {2 \left (-3 \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )+\sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )+5 \left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {\frac {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}\, \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right )}{3 d \left (5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\right )}^{\frac {3}{2}}}\)	\(485\)
default	\(\frac {\left (6 F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right )-E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right )+12 F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \cos \left (d x +c \right )-2 E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \cos \left (d x +c \right )+6 \sqrt {2}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )-\sqrt {2}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )-6 \sin \left (d x +c \right )\right ) \sqrt {3-2 \cos \left (d x +c \right )}}{3 d \left (-3+2 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right )}\)	\(519\)

[In]

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

[Out]

1/d*(1+cos(d*x+c))*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(3-2*cos(d*x+c))^(1/2)*(-2*(-3+2*cos(d*x+c))/(1+c
os(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),I*5^(1/2))/(-3+2*cos(d*x+c))/cos(d*x+c)^(1/2)-2/3/d*(-3*(-cs
c(d*x+c)^2*(1-cos(d*x+c))^2+1)^(1/2)*(5*csc(d*x+c)^2*(1-cos(d*x+c))^2+1)^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c)
,I*5^(1/2))+(-csc(d*x+c)^2*(1-cos(d*x+c))^2+1)^(1/2)*(5*csc(d*x+c)^2*(1-cos(d*x+c))^2+1)^(1/2)*EllipticE(cot(d
*x+c)-csc(d*x+c),I*5^(1/2))+5*csc(d*x+c)^3*(1-cos(d*x+c))^3+csc(d*x+c)-cot(d*x+c))*((5*csc(d*x+c)^2*(1-cos(d*x
+c))^2+1)/(csc(d*x+c)^2*(1-cos(d*x+c))^2+1))^(1/2)*(csc(d*x+c)^2*(1-cos(d*x+c))^2-1)/(5*csc(d*x+c)^2*(1-cos(d*
x+c))^2+1)/(csc(d*x+c)^2*(1-cos(d*x+c))^2+1)/(-(csc(d*x+c)^2*(1-cos(d*x+c))^2-1)/(csc(d*x+c)^2*(1-cos(d*x+c))^
2+1))^(3/2)

Fricas [F]

\[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(-(cos(d*x + c) + 1)*sqrt(-2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^3 - 3*cos(d*x + c)^2
), x)

Sympy [F]

\[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {3 - 2 \cos {\left (c + d x \right )}} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral((cos(c + d*x) + 1)/(sqrt(3 - 2*cos(c + d*x))*cos(c + d*x)**(3/2)), x)

Maxima [F]

\[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(-2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)), x)

Giac [F]

\[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((cos(d*x + c) + 1)/(sqrt(-2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\cos (c+d x)}{\sqrt {3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {3-2\,\cos \left (c+d\,x\right )}} \,d x \]

[In]

int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(3 - 2*cos(c + d*x))^(1/2)),x)

[Out]

int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(3 - 2*cos(c + d*x))^(1/2)), x)