Integrand size = 33, antiderivative size = 72 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \] Output:
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
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx \] Input:
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]) ,x]
Output:
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]) , x]
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {3042, 3473}
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.
| \(\displaystyle \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {2 \cos (c+d x)+3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {2 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\left .\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right )}{3 d}\) |
Input:
Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]),x]
Output:
(2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Co s[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (65 ) = 130\).
Time = 13.90 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.28
| method | result | size |
| default | \(\frac {\left (30+20 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (-6 \cos \left (d x +c \right )^{2}-12 \cos \left (d x +c \right )-6\right ) \sqrt {2}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (5 \cos \left (d x +c \right )^{2}+10 \cos \left (d x +c \right )+5\right ) \sqrt {2}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{15 d \sqrt {3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+1\right )}\) | \(236\) |
| parts | \(-\frac {\left (\left (-20 \cos \left (d x +c \right )-30\right ) \sin \left (d x +c \right )+\left (3 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+3\right ) \sqrt {2}\, \sqrt {10}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (-5 \cos \left (d x +c \right )^{2}-10 \cos \left (d x +c \right )-5\right ) \sqrt {2}\, \sqrt {10}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {3+2 \cos \left (d x +c \right )}}{15 d \sqrt {\cos \left (d x +c \right )}\, \left (2 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+3\right )}-\frac {\left (\cos \left (d x +c \right )+1\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )}{5 d \sqrt {\cos \left (d x +c \right )}\, \sqrt {3+2 \cos \left (d x +c \right )}}\) | \(352\) |
Input:
int((cos(d*x+c)+1)/cos(d*x+c)^(3/2)/(3+2*cos(d*x+c))^(1/2),x,method=_RETUR NVERBOSE)
Output:
1/15/d*((30+20*cos(d*x+c))*sin(d*x+c)+(-6*cos(d*x+c)^2-12*cos(d*x+c)-6)*2^ (1/2)*10^(1/2)*((3+2*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c )-csc(d*x+c),1/5*I*5^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+(5*cos(d*x+c )^2+10*cos(d*x+c)+5)*2^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(cos(d*x+c)+1))^(1 /2)*EllipticE(cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/2))*(cos(d*x+c)/(cos(d*x+c) +1))^(1/2))/(3+2*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)/(cos(d*x+c)+1)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(3+2*cos(d*x+c))^(1/2),x, algori thm="fricas")
Output:
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)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {2 \cos {\left (c + d x \right )} + 3} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)**(3/2)/(3+2*cos(d*x+c))**(1/2),x)
Output:
Integral((cos(c + d*x) + 1)/(sqrt(2*cos(c + d*x) + 3)*cos(c + d*x)**(3/2)) , x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(3+2*cos(d*x+c))^(1/2),x, algori thm="maxima")
Output:
integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)) , x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(3+2*cos(d*x+c))^(1/2),x, algori thm="giac")
Output:
integrate((cos(d*x + c) + 1)/(sqrt(2*cos(d*x + c) + 3)*cos(d*x + c)^(3/2)) , x)
Timed out. \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \] Input:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2*cos(c + d*x) + 3)^(1/2)),x)
Output:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2*cos(c + d*x) + 3)^(1/2)), x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {2 \cos \left (d x +c \right )+3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}}d x +\int \frac {\sqrt {2 \cos \left (d x +c \right )+3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )}d x \] Input:
int((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(3+2*cos(d*x+c))^(1/2),x)
Output:
int((sqrt(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x)))/(2*cos(c + d*x)**3 + 3*c os(c + d*x)**2),x) + int((sqrt(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x)))/(2* cos(c + d*x)**2 + 3*cos(c + d*x)),x)