Integrand size = 33, antiderivative size = 93 \[ \int \frac {1+\cos (c+d x)}{\sqrt {2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {5} \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \] Output:
5^(1/2)*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*csc(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+s ec(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 = 0.47 (sec) , antiderivative size = 93, 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.
| \(\displaystyle \int \frac {\cos (c+d x)+1}{\sqrt {2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3474 |
| \(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\cos (c+d x)+1}{\sqrt {2-3 \cos (c+d x)} (-\cos (c+d x))^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {\sqrt {5} \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} E\left (\arcsin \left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right )}{d}\) |
Input:
Int[(1 + Cos[c + d*x])/(Sqrt[2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]
Output:
(Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[Arc Sin[Sqrt[2 - 3*Cos[c + d*x]]/Sqrt[-Cos[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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(80)=160\).
Time = 13.57 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(\frac {\left (\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (4 \cos \left (d x +c \right )^{2}+8 \cos \left (d x +c \right )+4\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )\right ) \sqrt {2-3 \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )-2\right )}\) | \(225\) |
| parts | \(\frac {\left (\left (-2+3 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (2+2 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )\right ) \sqrt {2-3 \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )-2\right )}-\frac {2 \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(320\) |
Input:
int((cos(d*x+c)+1)/(2-3*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x,method=_RETUR NVERBOSE)
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)*EllipticE (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)/( 3*cos(d*x+c)^2+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, algori thm="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)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {2 - 3 \cos {\left (c + d x \right )}} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] 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(2 - 3*cos(c + d*x))*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, algori thm="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, algori thm="giac")
Output:
integrate((cos(d*x + c) + 1)/(sqrt(-3*cos(d*x + c) + 2)*cos(d*x + c)^(3/2) ), x)
Timed out. \[ \int \frac {1+\cos (c+d x)}{\sqrt {2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \] Input:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2 - 3*cos(c + d*x))^(1/2)),x)
Output:
int((cos(c + d*x) + 1)/(cos(c + d*x)^(3/2)*(2 - 3*cos(c + d*x))^(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))