Integrand size = 21, antiderivative size = 111 \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f} \]
-7/20*cos(f*x+e)*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(c os(e+1/4*Pi+f*x),2^(1/2))*(d*tan(f*x+e))^(1/2)/f/sin(2*f*x+2*e)^(1/2)+7/30 *cos(f*x+e)^3*(d*tan(f*x+e))^(3/2)/d/f+1/5*cos(f*x+e)^5*(d*tan(f*x+e))^(3/ 2)/d/f
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.83 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \left (20 \sin (2 (e+f x))+3 \sin (4 (e+f x))+28 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x)} \tan (e+f x)\right )}{120 f} \]
(Cos[e + f*x]*Sqrt[d*Tan[e + f*x]]*(20*Sin[2*(e + f*x)] + 3*Sin[4*(e + f*x )] + 28*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[e + f*x]^2]*Sqrt[Sec[e + f*x ]^2]*Tan[e + f*x]))/(120*f)
Time = 0.61 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3092, 3042, 3092, 3042, 3095, 3042, 3052, 3042, 3119}
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.
| \(\displaystyle \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {d \tan (e+f x)}}{\sec (e+f x)^5}dx\) |
| \(\Big \downarrow \) 3092 |
| \(\displaystyle \frac {7}{10} \int \cos ^3(e+f x) \sqrt {d \tan (e+f x)}dx+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \int \frac {\sqrt {d \tan (e+f x)}}{\sec (e+f x)^3}dx+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3092 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \int \cos (e+f x) \sqrt {d \tan (e+f x)}dx+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {1}{2} \int \frac {\sqrt {d \tan (e+f x)}}{\sec (e+f x)}dx+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3095 |
| \(\displaystyle \frac {7}{10} \left (\frac {\sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)} \int \sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}dx}{2 \sqrt {\sin (e+f x)}}+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {\sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)} \int \sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}dx}{2 \sqrt {\sin (e+f x)}}+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {7}{10} \left (\frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{2 \sqrt {\sin (2 e+2 f x)}}+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{10} \left (\frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{2 \sqrt {\sin (2 e+2 f x)}}+\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}\right )+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{10} \left (\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}+\frac {\cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{2 f \sqrt {\sin (2 e+2 f x)}}\right )\) |
(Cos[e + f*x]^5*(d*Tan[e + f*x])^(3/2))/(5*d*f) + (7*((Cos[e + f*x]*Ellipt icE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(2*f*Sqrt[Sin[2*e + 2*f*x]]) + (Cos[e + f*x]^3*(d*Tan[e + f*x])^(3/2))/(3*d*f)))/10
3.3.35.3.1 Defintions of rubi rules used
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(a*Sec[e + f*x])^m)*((b*Tan[e + f*x])^(n + 1)/(b*f* m)), x] + Simp[(m + n + 1)/(a^2*m) Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1 ] && EqQ[n, -2^(-1)])) && IntegersQ[2*m, 2*n]
Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[ Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(122)=244\).
Time = 1.38 (sec) , antiderivative size = 404, normalized size of antiderivative = 3.64
| method | result | size |
| default | \(-\frac {\csc \left (f x +e \right ) \left (12 \left (\cos ^{6}\left (f x +e \right )\right ) \sqrt {2}+2 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}+42 \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \cos \left (f x +e \right )-21 \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+42 \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}-21 \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+7 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right )-21 \sqrt {2}\, \cos \left (f x +e \right )\right ) \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}}{120 f}\) | \(404\) |
-1/120/f*csc(f*x+e)*(12*cos(f*x+e)^6*2^(1/2)+2*cos(f*x+e)^4*2^(1/2)+42*(co t(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticE((-cot (f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)* cos(f*x+e)-21*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2 )*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1 /2),1/2*2^(1/2))*cos(f*x+e)+42*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e) -csc(f*x+e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2)) *(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-21*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot( f*x+e)-csc(f*x+e))^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*EllipticF((-cot( f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))+7*2^(1/2)*cos(f*x+e)^2-21*2^(1/2)* cos(f*x+e))*(d*tan(f*x+e))^(1/2)*2^(1/2)
\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \]
Timed out. \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\text {Timed out} \]
\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \]
\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \]
Timed out. \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int {\cos \left (e+f\,x\right )}^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \]