\(\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx\) [445]

Optimal result

Integrand size = 33, antiderivative size = 72 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 72, 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} \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\left .\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right )}{3 d} \]

[In]

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

[Out]

(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -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 \cot (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (65 ) = 130\).

Time = 12.36 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.62

method	result	size
parts	\(-\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}+25}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )+5 \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}+25}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )+5 \left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3}+25 \csc \left (d x +c \right )-25 \cot \left (d x +c \right )\right ) \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+5}{\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 )}{15 d \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+5\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}}}-\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 {10}\, \sqrt {\frac {3+2 \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 ), \frac {i \sqrt {5}}{5}\right )}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\)	\(477\)
default	\(-\frac {6 \sqrt {2}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \left (\cos ^{2}\left (d x +c \right )\right )-5 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+12 \sqrt {2}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \cos \left (d x +c \right )-10 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \cos \left (d x +c \right )+6 \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \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 ), \frac {i \sqrt {5}}{5}\right ) \sqrt {2}-5 \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \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 ), \frac {i \sqrt {5}}{5}\right ) \sqrt {2}-20 \cos \left (d x +c \right ) \sin \left (d x +c \right )-30 \sin \left (d x +c \right )}{15 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \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]

-2/15/d*(-3*(-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+25)^(1/2)*EllipticF(cot(
d*x+c)-csc(d*x+c),1/5*I*5^(1/2))+5*(-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+2
5)^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/2))+5*csc(d*x+c)^3*(1-cos(d*x+c))^3+25*csc(d*x+c)-25*cot(d
*x+c))*((csc(d*x+c)^2*(1-cos(d*x+c))^2+5)/(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)/(csc(d*x+c)^2*(1-cos(d*x+c))^2+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))^(3/2)-1/5/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)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/
2))/cos(d*x+c)^(1/2)

Fricas [F]

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

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

Sympy [F]

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

[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(2*cos(c + d*x) + 3)*cos(c + d*x)**(3/2)), x)

Maxima [F]

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

[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]

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

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

[In]

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

[Out]

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