Integrand size = 33, antiderivative size = 96 \[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 d} \]
-2/3*csc(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))*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(1-sec(d*x+c))^(1/ 2)*(1+sec(d*x+c))^(1/2)/d
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \]
Time = 0.45 (sec) , antiderivative size = 96, 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.
| \(\displaystyle \int \frac {\cos (c+d x)+1}{\sqrt {-2 \cos (c+d x)-3} \cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {-2 \sin \left (c+d x+\frac {\pi }{2}\right )-3} \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3474 |
| \(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\cos (c+d x)+1}{\sqrt {-2 \cos (c+d x)-3} (-\cos (c+d x))^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {-\cos (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {-2 \sin \left (c+d x+\frac {\pi }{2}\right )-3} \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} E\left (\left .\arcsin \left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{3 d}\) |
(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[S qrt[-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)
3.5.48.3.1 Defintions of rubi rules used
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (85 ) = 170\).
Time = 12.62 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.45
| method | result | size |
| default | \(\frac {\left (i \sqrt {10}\, \sqrt {2}\, \sqrt {5}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, E\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+2 i \sqrt {10}\, \sqrt {2}\, \sqrt {5}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, E\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right ) \cos \left (d x +c \right )+i \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, E\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}-20 \cos \left (d x +c \right ) \sin \left (d x +c \right )-30 \sin \left (d x +c \right )\right ) \sqrt {-3-2 \cos \left (d x +c \right )}}{15 d \left (3+2 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right )}\) | \(331\) |
| parts | \(\frac {i \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 {-3-2 \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 (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right ) \sqrt {5}}{5 d \left (3+2 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}-\frac {2 \left (i \sqrt {5}\, \sqrt {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+25}\, \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, E\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right )-3 i \sqrt {5}\, \sqrt {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+25}\, \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, F\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {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}}}\) | \(520\) |
1/15/d*(I*10^(1/2)*2^(1/2)*5^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2) *(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(1/5*I*(csc(d*x+c)-cot(d*x+c)) *5^(1/2),I*5^(1/2))*cos(d*x+c)^2+2*I*10^(1/2)*2^(1/2)*5^(1/2)*((3+2*cos(d* x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(1/ 5*I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),I*5^(1/2))*cos(d*x+c)+I*(cos(d*x+c)/(1 +cos(d*x+c)))^(1/2)*EllipticE(1/5*I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),I*5^(1 /2))*5^(1/2)*2^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*10^(1/2)-20*c os(d*x+c)*sin(d*x+c)-30*sin(d*x+c))*(-3-2*cos(d*x+c))^(1/2)/(3+2*cos(d*x+c ))/cos(d*x+c)^(1/2)/(1+cos(d*x+c))
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
integral(-(cos(d*x + c) + 1)*sqrt(-2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))/ (2*cos(d*x + c)^3 + 3*cos(d*x + c)^2), x)
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {- 2 \cos {\left (c + d x \right )} - 3} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1+\cos (c+d x)}{\sqrt {-3-2 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {-2\,\cos \left (c+d\,x\right )-3}} \,d x \]