Integrand size = 21, antiderivative size = 109 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {5 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {d \tan (e+f x)}}+\frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f} \]
-5/12*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi +f*x),2^(1/2))*sec(f*x+e)*sin(2*f*x+2*e)^(1/2)/f/(d*tan(f*x+e))^(1/2)+5/6* cos(f*x+e)*(d*tan(f*x+e))^(1/2)/d/f+1/3*cos(f*x+e)^3*(d*tan(f*x+e))^(1/2)/ d/f
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {11 \sin (e+f x)+\sin (3 (e+f x))-10 \sqrt [4]{-1} \cos (e+f x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (e+f x)}\right ),-1\right ) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}}{12 f \sqrt {d \tan (e+f x)}} \]
(11*Sin[e + f*x] + Sin[3*(e + f*x)] - 10*(-1)^(1/4)*Cos[e + f*x]*EllipticF [I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[e + f*x]]], -1]*Sqrt[Sec[e + f*x]^2]*Sqrt[T an[e + f*x]])/(12*f*Sqrt[d*Tan[e + f*x]])
Time = 0.58 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.02, 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, 3094, 3042, 3053, 3042, 3120}
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.
\(\displaystyle \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sec (e+f x)^3 \sqrt {d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 3092 |
\(\displaystyle \frac {5}{6} \int \frac {\cos (e+f x)}{\sqrt {d \tan (e+f x)}}dx+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \int \frac {1}{\sec (e+f x) \sqrt {d \tan (e+f x)}}dx+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3092 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}}dx+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {1}{2} \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}}dx+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3094 |
\(\displaystyle \frac {5}{6} \left (\frac {\sqrt {\sin (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}}dx}{2 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}}+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {\sqrt {\sin (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}}dx}{2 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}}+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {5}{6} \left (\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \tan (e+f x)}}+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {d \tan (e+f x)}}+\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}\right )+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {5}{6} \left (\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}+\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \tan (e+f x)}}\right )\) |
(Cos[e + f*x]^3*Sqrt[d*Tan[e + f*x]])/(3*d*f) + (5*((EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[d*Tan[e + f*x]]) + (Cos[e + f*x]*Sqrt[d*Tan[e + f*x]])/(d*f)))/6
3.3.57.3.1 Defintions of rubi rules used
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/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[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Result contains complex when optimal does not.
Time = 6.23 (sec) , antiderivative size = 1906, normalized size of antiderivative = 17.49
-1/48/f/(d*tan(f*x+e))^(1/2)*(6*I*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x +e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2* I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-6*I*sec(f*x+e)*(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)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2 ))-6*I*(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)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/ 2+1/2*I,1/2*2^(1/2))+6*I*sec(f*x+e)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f *x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/ 2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)+6*(cot(f*x+e)-csc(f*x+e) +1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e) +1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)-32*(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))+6* (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)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I ,1/2*2^(1/2))-8*cos(f*x+e)^2*sin(f*x+e)*2^(1/2)+6*sec(f*x+e)*(cot(f*x+e)-c sc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+c sc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2) -32*sec(f*x+e)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+...
\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^3}{\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]