3.5.12 \(\int \sqrt {\sec ^4(x) \tan (x)} \, dx\) [412]

3.5.12.1 Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\frac {2}{3} \cos (x) \sin (x) \sqrt {\sec ^4(x) \tan (x)} \]

output

2/3*cos(x)*sin(x)*(sec(x)^4*tan(x))^(1/2)
 

3.5.12.2 Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\frac {2}{3} \cos (x) \sin (x) \sqrt {\sec ^4(x) \tan (x)} \]

input

Integrate[Sqrt[Sec[x]^4*Tan[x]],x]
 

output

(2*Cos[x]*Sin[x]*Sqrt[Sec[x]^4*Tan[x]])/3
 

3.5.12.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4889, 2096, 2001, 1383, 15}

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.

\(\displaystyle \int \sqrt {\tan (x) \sec ^4(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {\tan (x) \sec (x)^4}dx\)

\(\Big \downarrow \) 4889

\(\displaystyle \int \frac {\tan (x) \left (\tan ^2(x)+1\right )}{\sqrt {\tan (x) \left (\tan ^2(x)+1\right )^2}}d\tan (x)\)

\(\Big \downarrow \) 2096

\(\displaystyle \int \frac {\tan (x) \left (\tan ^2(x)+1\right )}{\sqrt {\tan ^5(x)+2 \tan ^3(x)+\tan (x)}}d\tan (x)\)

\(\Big \downarrow \) 2001

\(\displaystyle \frac {\sqrt {\tan (x)} \sqrt {\tan ^4(x)+2 \tan ^2(x)+1} \int \frac {\sqrt {\tan (x)} \left (\tan ^2(x)+1\right )}{\sqrt {\tan ^4(x)+2 \tan ^2(x)+1}}d\tan (x)}{\sqrt {\tan ^5(x)+2 \tan ^3(x)+\tan (x)}}\)

\(\Big \downarrow \) 1383

\(\displaystyle \frac {\sqrt {\tan (x)} \left (\tan ^2(x)+1\right ) \int \sqrt {\tan (x)}d\tan (x)}{\sqrt {\tan ^5(x)+2 \tan ^3(x)+\tan (x)}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2 \tan ^2(x) \left (\tan ^2(x)+1\right )}{3 \sqrt {\tan ^5(x)+2 \tan ^3(x)+\tan (x)}}\)

input

Int[Sqrt[Sec[x]^4*Tan[x]],x]
 

output

(2*Tan[x]^2*(1 + Tan[x]^2))/(3*Sqrt[Tan[x] + 2*Tan[x]^3 + Tan[x]^5])
 

3.5.12.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 1383

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^(2 
*p)   Int[u*(d + e*x^n)^(q + 2*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q} 
, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&  !Int 
egerQ[p]
 

rule 2001

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(k_.) + (c_.)*(x_)^(n_.))^(p_ 
)*((A_) + (B_.)*(x_)^(q_)), x_Symbol] :> Simp[(a*x^j + b*x^k + c*x^n)^p/(x^ 
(j*p)*(a + b*x^(k - j) + c*x^(2*(k - j)))^p)   Int[x^(m + j*p)*(A + B*x^(k 
- j))*(a + b*x^(k - j) + c*x^(2*(k - j)))^p, x], x] /; FreeQ[{a, b, c, A, B 
, j, k, m, p}, x] && EqQ[q, k - j] && EqQ[n, 2*k - j] &&  !IntegerQ[p] && P 
osQ[k - j]
 

rule 2096

Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_), x_Symbol] :> Int[(f*x)^m*ExpandToSu 
m[z, x]*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p}, x] && BinomialQ[z, x] & 
& GeneralizedTrinomialQ[u, x] && EqQ[BinomialDegree[z, x] - GeneralizedTrin 
omialDegree[u, x], 0] &&  !(BinomialMatchQ[z, x] && GeneralizedTrinomialMat 
chQ[u, x])
 

rule 3042

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

rule 4889

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors 
[Tan[v], x]}, Simp[d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] &&  !MatchQ[ 
u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I 
ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
 

3.5.12.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {2 \cos \left (x \right ) \sin \left (x \right ) \sqrt {\left (\sec ^{4}\left (x \right )\right ) \tan \left (x \right )}}{3}\)	\(16\)

input

int((sin(x)/cos(x)^5)^(1/2),x,method=_RETURNVERBOSE)
 

output

2/3*cos(x)*sin(x)*(sec(x)^4*tan(x))^(1/2)
 

3.5.12.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\frac {2}{3} \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right )^{5}}} \cos \left (x\right ) \sin \left (x\right ) \]

input

integrate((sin(x)/cos(x)^5)^(1/2),x, algorithm="fricas")
 

output

2/3*sqrt(sin(x)/cos(x)^5)*cos(x)*sin(x)
 

3.5.12.6 Sympy [F(-1)]

Timed out. \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\text {Timed out} \]

input

integrate((sin(x)/cos(x)**5)**(1/2),x)
 

output

Timed out
 

3.5.12.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.32 \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\frac {2}{3} \, \tan \left (x\right )^{\frac {3}{2}} \]

input

integrate((sin(x)/cos(x)^5)^(1/2),x, algorithm="maxima")
 

output

2/3*tan(x)^(3/2)
 

3.5.12.8 Giac [F]

\[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\int { \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right )^{5}}} \,d x } \]

input

integrate((sin(x)/cos(x)^5)^(1/2),x, algorithm="giac")
 

output

integrate(sqrt(sin(x)/cos(x)^5), x)
 

3.5.12.9 Mupad [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\frac {\sin \left (2\,x\right )\,\sqrt {\frac {\sin \left (x\right )}{{\cos \left (x\right )}^5}}}{3} \]

input

int((sin(x)/cos(x)^5)^(1/2),x)
 

output

(sin(2*x)*(sin(x)/cos(x)^5)^(1/2))/3
 

3.5.12.10 Reduce [F]

\[ \int \sqrt {\sec ^4(x) \tan (x)} \, dx=\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}}{\cos \left (x \right )^{3}}d x \]

input

int(sqrt(sin(x)/cos(x))/cos(x)**2,x)
 

output

int((sqrt(sin(x))*sqrt(cos(x)))/cos(x)**3,x)