\(\int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\) [257]

Optimal result

Integrand size = 21, antiderivative size = 109 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {5 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {d \tan (e+f x)}}+\frac {5 \cos (e+f x) \sqrt {d \tan (e+f x)}}{6 d f}+\frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f} \] Output:

5/12*InverseJacobiAM(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
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {11 \sin (e+f x)+\sin (3 (e+f x))-10 \sqrt [4]{-1} \cos (e+f x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (e+f x)}\right ),-1\right ) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}}{12 f \sqrt {d \tan (e+f x)}} \] Input:

Integrate[Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]
 

Output:

(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]])
 

Rubi [A] (verified)

Time = 0.56 (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

\(\displaystyle \frac {\cos ^3(e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {5}{6} \left (\frac {\cos (e+f x) \sqrt {d \tan (e+f x)}}{d f}+\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{2 f \sqrt {d \tan (e+f x)}}\right )\)

Input:

Int[Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]],x]
 

Output:

(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
 

Defintions of rubi rules used

rule 3042

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

rule 3053

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]
 

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 3094

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]
 

rule 3120

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.77 (sec) , antiderivative size = 1507, normalized size of antiderivative = 13.83

method	result	size
default	\(\text {Expression too large to display}\)	\(1507\)

Input:

int(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/96/f/(d*tan(f*x+e))^(1/2)*(2^(1/2)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x 
+e))^2)^(1/2)*ln(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-2*sin 
(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+csc(f*x+e)-sin(f*x+e)-2*cos(f*x 
+e)+2)/(cos(f*x+e)-1))*(sin(f*x+e)*(6*cos(f*x+e)+6)-3-3*sec(f*x+e))+sin(f* 
x+e)*(-12*cos(f*x+e)-12)*(-sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*l 
n(-(cot(f*x+e)*cos(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x 
+e)/(1+cos(f*x+e))^2)^(1/2)+csc(f*x+e)-sin(f*x+e)-2*cos(f*x+e)+2)/(cos(f*x 
+e)-1))+2^(1/2)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*ln((2*si 
n(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cot(f*x+e)*cos( 
f*x+e)+sin(f*x+e)+2*cos(f*x+e)+2*cot(f*x+e)-csc(f*x+e)-2)/(cos(f*x+e)-1))* 
(sin(f*x+e)*(-6*cos(f*x+e)-6)+3+3*sec(f*x+e))+sin(f*x+e)*(12*cos(f*x+e)+12 
)*(-sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*ln((2*sin(f*x+e)*(-2*sin 
(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cot(f*x+e)*cos(f*x+e)+sin(f*x+e 
)+2*cos(f*x+e)+2*cot(f*x+e)-csc(f*x+e)-2)/(cos(f*x+e)-1))+2^(1/2)*(-2*sin( 
f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*arctan((sin(f*x+e)*(-2*sin(f*x+e 
)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)-1))*(sin(f* 
x+e)*(12*cos(f*x+e)+12)-6-6*sec(f*x+e))+sin(f*x+e)*(-24*cos(f*x+e)-24)*(-s 
in(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*arctan((sin(f*x+e)*(-2*sin(f* 
x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)-1))+2^(1 
/2)*(-2*sin(f*x+e)*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*arctan((sin(f*x+e...
 

Fricas [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(d*tan(f*x + e))*cos(f*x + e)^3/(d*tan(f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**3/(d*tan(f*x+e))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(cos(f*x + e)^3/sqrt(d*tan(f*x + e)), x)
 

Giac [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(cos(f*x + e)^3/sqrt(d*tan(f*x + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^3}{\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \] Input:

int(cos(e + f*x)^3/(d*tan(e + f*x))^(1/2),x)
 

Output:

int(cos(e + f*x)^3/(d*tan(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right )}\, \cos \left (f x +e \right )^{3}}{\tan \left (f x +e \right )}d x \right )}{d} \] Input:

int(cos(f*x+e)^3/(d*tan(f*x+e))^(1/2),x)
 

Output:

(sqrt(d)*int((sqrt(tan(e + f*x))*cos(e + f*x)**3)/tan(e + f*x),x))/d