\(\int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx\) [235]

Optimal result

Integrand size = 21, antiderivative size = 111 \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f} \] Output:

-7/20*cos(f*x+e)*EllipticE(cos(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
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.77 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \left (20 \sin (2 (e+f x))+3 \sin (4 (e+f x))+28 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x)} \tan (e+f x)\right )}{120 f} \] Input:

Integrate[Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]],x]
 

Output:

(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)
 

Rubi [A] (verified)

Time = 0.58 (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

\(\displaystyle \frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{10} \left (\frac {\cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{3 d f}+\frac {\cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{2 f \sqrt {\sin (2 e+2 f x)}}\right )\)

Input:

Int[Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]],x]
 

Output:

(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
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3052

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]
 

rule 3092

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]
 

rule 3095

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]
 

rule 3119

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(98)=196\).

Time = 1.48 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.25

method	result	size
default	\(\frac {\left (-\frac {\sqrt {2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \left (-21 \cot \left (f x +e \right )-21 \csc \left (f x +e \right )\right )}{120}-\frac {\sqrt {2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (42 \cot \left (f x +e \right )+42 \csc \left (f x +e \right )\right )}{120}-\frac {\cot \left (f x +e \right ) \left (24 \cos \left (f x +e \right )^{5}+4 \cos \left (f x +e \right )^{3}+14 \cos \left (f x +e \right )-42\right )}{120}\right ) \sqrt {d \tan \left (f x +e \right )}}{f}\)	\(250\)

Input:

int(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/f*(-1/120*(2*cot(f*x+e)-2*csc(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^( 
1/2)*EllipticF((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*(csc(f*x+e)-co 
t(f*x+e)+1)^(1/2)*(-21*cot(f*x+e)-21*csc(f*x+e))-1/120*(2*cot(f*x+e)-2*csc 
(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(csc(f*x+e)-cot(f*x+e)+1)^ 
(1/2)*EllipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*(42*cot(f*x+e 
)+42*csc(f*x+e))-1/120*cot(f*x+e)*(24*cos(f*x+e)^5+4*cos(f*x+e)^3+14*cos(f 
*x+e)-42))*(d*tan(f*x+e))^(1/2)
 

Fricas [F]

\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**5*(d*tan(f*x+e))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)
 

Giac [F]

\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int { \sqrt {d \tan \left (f x + e\right )} \cos \left (f x + e\right )^{5} \,d x } \] Input:

integrate(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*tan(f*x + e))*cos(f*x + e)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\int {\cos \left (e+f\,x\right )}^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \] Input:

int(cos(e + f*x)^5*(d*tan(e + f*x))^(1/2),x)
 

Output:

int(cos(e + f*x)^5*(d*tan(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (f x +e \right )}\, \cos \left (f x +e \right )^{5}d x \right ) \] Input:

int(cos(f*x+e)^5*(d*tan(f*x+e))^(1/2),x)
 

Output:

sqrt(d)*int(sqrt(tan(e + f*x))*cos(e + f*x)**5,x)