Integrand size = 33, antiderivative size = 95 \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{d} \] Output:
(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE(1/5*(-2-3*cos(d* x+c))^(1/2)*5^(1/2)/(-cos(d*x+c))^(1/2),5^(1/2))*(-1-sec(d*x+c))^(1/2)*(1- sec(d*x+c))^(1/2)/d
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \] Input:
Integrate[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2) ),x]
Output:
Integrate[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2) ), x]
Time = 0.46 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 3474, 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.
\(\displaystyle \int \frac {\cos (c+d x)+1}{\sqrt {-3 \cos (c+d x)-2} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2} \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3474 |
\(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\cos (c+d x)+1}{\sqrt {-3 \cos (c+d x)-2} (-\cos (c+d x))^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2} \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{d}\) |
Input:
Int[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]
Output:
(Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt [-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 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]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-Sqrt [(-b)*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]] Int[(A + B*Sin[e + f*x])/(((-b)* Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(c + d)/b]
Time = 13.94 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {\left (\left (-30 \cos \left (d x +c \right )-20\right ) \sin \left (d x +c \right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {10}\, \sqrt {5}\, \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-2-3 \cos \left (d x +c \right )}}{10 d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+2\right )}\) | \(160\) |
parts | \(\frac {\left (\left (-30 \cos \left (d x +c \right )-20\right ) \sin \left (d x +c \right )+\left (2+2 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )\right ) \sqrt {10}\, \sqrt {5}\, \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {10}\, \sqrt {5}\, \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-2-3 \cos \left (d x +c \right )}}{10 d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+2\right )}+\frac {\sqrt {2}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {5}}{5 d \sqrt {-2-3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(365\) |
Input:
int((cos(d*x+c)+1)/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x,method=_RETU RNVERBOSE)
Output:
1/10/d*((-30*cos(d*x+c)-20)*sin(d*x+c)+(cos(d*x+c)^2+2*cos(d*x+c)+1)*10^(1 /2)*5^(1/2)*2^(1/2)*EllipticE(1/5*(csc(d*x+c)-cot(d*x+c))*5^(1/2),5^(1/2)) *(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2) )*(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)/(3*cos(d*x+c)^2+5*cos(d*x+c)+2)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algor ithm="fricas")
Output:
integral(-(cos(d*x + c) + 1)*sqrt(-3*cos(d*x + c) - 2)*sqrt(cos(d*x + c))/ (3*cos(d*x + c)^3 + 2*cos(d*x + c)^2), x)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((1+cos(d*x+c))/(-2-3*cos(d*x+c))**(1/2)/cos(d*x+c)**(3/2),x)
Output:
Integral((cos(c + d*x) + 1)/(sqrt(-3*cos(c + d*x) - 2)*cos(c + d*x)**(3/2) ), x)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),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)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((1+cos(d*x+c))/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),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. \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \] 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)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=-\left (\int \frac {\sqrt {-3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{3}+2 \cos \left (d x +c \right )^{2}}d x \right )-\left (\int \frac {\sqrt {-3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )}d x \right ) \] Input:
int((1+cos(d*x+c))/(-2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x)
Output:
- (int((sqrt( - 3*cos(c + d*x) - 2)*sqrt(cos(c + d*x)))/(3*cos(c + d*x)** 3 + 2*cos(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))