Integrand size = 33, antiderivative size = 70 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=-\frac {\sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \] Output:
-5^(1/2)*cot(d*x+c)*EllipticE((-2+3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),1/5 *5^(1/2))*(-1+sec(d*x+c))^(1/2)*(1+sec(d*x+c))^(1/2)/d
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx \] Input:
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]] ),x]
Output:
Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]] ), x]
Time = 0.30 (sec) , antiderivative size = 70, 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.
\(\displaystyle \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {3 \cos (c+d x)-2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle -\frac {\sqrt {5} \cot (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} E\left (\arcsin \left (\frac {\sqrt {3 \cos (c+d x)-2}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right )}{d}\) |
Input:
Int[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]]),x]
Output:
-((Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Co s[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/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]
Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(61)=122\).
Time = 11.94 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.09
method | result | size |
default | \(-\frac {\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (4 \cos \left (d x +c \right )^{2}+8 \cos \left (d x +c \right )+4\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, \left (\cos \left (d x +c \right )+1\right )}\) | \(216\) |
parts | \(-\frac {\left (\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (2+2 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )\right ) \sqrt {-2+3 \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )-2\right )}-\frac {2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {\cos \left (d x +c \right )}\, \sqrt {-2+3 \cos \left (d x +c \right )}}\) | \(321\) |
Input:
int((cos(d*x+c)+1)/cos(d*x+c)^(3/2)/(-2+3*cos(d*x+c))^(1/2),x,method=_RETU RNVERBOSE)
Output:
-1/d*((-2+3*cos(d*x+c))*sin(d*x+c)+(4*cos(d*x+c)^2+8*cos(d*x+c)+4)*(cos(d* x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),5^(1/2))*((-2+3 *cos(d*x+c))/(cos(d*x+c)+1))^(1/2)+(cos(d*x+c)^2+2*cos(d*x+c)+1)*(cos(d*x+ c)/(cos(d*x+c)+1))^(1/2)*((-2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elliptic E(cot(d*x+c)-csc(d*x+c),5^(1/2)))/(-2+3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2) /(cos(d*x+c)+1)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-2+3*cos(d*x+c))^(1/2),x, algor ithm="fricas")
Output:
integral(sqrt(3*cos(d*x + c) - 2)*(cos(d*x + c) + 1)*sqrt(cos(d*x + c))/(3 *cos(d*x + c)^3 - 2*cos(d*x + c)^2), x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {3 \cos {\left (c + d x \right )} - 2} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)**(3/2)/(-2+3*cos(d*x+c))**(1/2),x)
Output:
Integral((cos(c + d*x) + 1)/(sqrt(3*cos(c + d*x) - 2)*cos(c + d*x)**(3/2)) , x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-2+3*cos(d*x+c))^(1/2),x, algor ithm="maxima")
Output:
integrate((cos(d*x + c) + 1)/(sqrt(3*cos(d*x + c) - 2)*cos(d*x + c)^(3/2)) , x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-2+3*cos(d*x+c))^(1/2),x, algor ithm="giac")
Output:
integrate((cos(d*x + c) + 1)/(sqrt(3*cos(d*x + c) - 2)*cos(d*x + c)^(3/2)) , x)
Timed out. \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {3\,\cos \left (c+d\,x\right )-2}} \,d x \] Input:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(3*cos(c + d*x) - 2)^(1/2)),x)
Output:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(3*cos(c + d*x) - 2)^(1/2)), x)
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{3}-2 \cos \left (d x +c \right )^{2}}d x +\int \frac {\sqrt {3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )}d x \] Input:
int((1+cos(d*x+c))/cos(d*x+c)^(3/2)/(-2+3*cos(d*x+c))^(1/2),x)
Output:
int((sqrt(3*cos(c + d*x) - 2)*sqrt(cos(c + d*x)))/(3*cos(c + d*x)**3 - 2*c os(c + d*x)**2),x) + int((sqrt(3*cos(c + d*x) - 2)*sqrt(cos(c + d*x)))/(3* cos(c + d*x)**2 - 2*cos(c + d*x)),x)