\(\int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx\) [378]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\frac {80 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{77 f}-\frac {40 b \sin (e+f x)}{77 f \sqrt {b \sec (e+f x)}}-\frac {20 b \sin ^3(e+f x)}{77 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}} \] Output:

80/77*cos(f*x+e)^(1/2)*InverseJacobiAM(1/2*f*x+1/2*e,2^(1/2))*(b*sec(f*x+e 
))^(1/2)/f-40/77*b*sin(f*x+e)/f/(b*sec(f*x+e))^(1/2)-20/77*b*sin(f*x+e)^3/ 
f/(b*sec(f*x+e))^(1/2)-2/11*b*sin(f*x+e)^5/f/(b*sec(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59 \[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\frac {\sqrt {b \sec (e+f x)} \left (1280 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )-435 \sin (2 (e+f x))+68 \sin (4 (e+f x))-7 \sin (6 (e+f x))\right )}{1232 f} \] Input:

Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^6,x]
 

Output:

(Sqrt[b*Sec[e + f*x]]*(1280*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 
 435*Sin[2*(e + f*x)] + 68*Sin[4*(e + f*x)] - 7*Sin[6*(e + f*x)]))/(1232*f 
)
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3107, 3042, 3107, 3042, 3107, 3042, 4258, 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.

\(\displaystyle \int \sin ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {b \sec (e+f x)}}{\csc (e+f x)^6}dx\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {10}{11} \int \sqrt {b \sec (e+f x)} \sin ^4(e+f x)dx-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10}{11} \int \frac {\sqrt {b \sec (e+f x)}}{\csc (e+f x)^4}dx-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \int \sqrt {b \sec (e+f x)} \sin ^2(e+f x)dx-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \int \frac {\sqrt {b \sec (e+f x)}}{\csc (e+f x)^2}dx-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \left (\frac {2}{3} \int \sqrt {b \sec (e+f x)}dx-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \left (\frac {2}{3} \int \sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \left (\frac {2}{3} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \left (\frac {2}{3} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {10}{11} \left (\frac {6}{7} \left (\frac {4 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{3 f}-\frac {2 b \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\right )-\frac {2 b \sin ^5(e+f x)}{11 f \sqrt {b \sec (e+f x)}}\)

Input:

Int[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^6,x]
 

Output:

(-2*b*Sin[e + f*x]^5)/(11*f*Sqrt[b*Sec[e + f*x]]) + (10*((-2*b*Sin[e + f*x 
]^3)/(7*f*Sqrt[b*Sec[e + f*x]]) + (6*((4*Sqrt[Cos[e + f*x]]*EllipticF[(e + 
 f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*f) - (2*b*Sin[e + f*x])/(3*f*Sqrt[b*S 
ec[e + f*x]])))/7))/11
 

Defintions of rubi rules used

rule 3042

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

rule 3107

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_.), x_Symbol] :> Simp[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1) 
/(a*f*(m + n))), x] + Simp[(m + 1)/(a^2*(m + n))   Int[(a*Csc[e + f*x])^(m 
+ 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] 
&& NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

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]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 12.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right ) \cos \left (f x +e \right ) \left (-7 \cos \left (f x +e \right )^{4}+24 \cos \left (f x +e \right )^{2}-37\right )+i \left (40 \cos \left (f x +e \right )+40\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )\right ) \sqrt {b \sec \left (f x +e \right )}}{77 f}\)	\(117\)

Input:

int((b*sec(f*x+e))^(1/2)*sin(f*x+e)^6,x,method=_RETURNVERBOSE)
 

Output:

2/77/f*(sin(f*x+e)*cos(f*x+e)*(-7*cos(f*x+e)^4+24*cos(f*x+e)^2-37)+I*(40*c 
os(f*x+e)+40)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*E 
llipticF(I*(cot(f*x+e)-csc(f*x+e)),I))*(b*sec(f*x+e))^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.86 \[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=-\frac {2 \, {\left ({\left (7 \, \cos \left (f x + e\right )^{5} - 24 \, \cos \left (f x + e\right )^{3} + 37 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 20 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 20 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{77 \, f} \] Input:

integrate((b*sec(f*x+e))^(1/2)*sin(f*x+e)^6,x, algorithm="fricas")
 

Output:

-2/77*((7*cos(f*x + e)^5 - 24*cos(f*x + e)^3 + 37*cos(f*x + e))*sqrt(b/cos 
(f*x + e))*sin(f*x + e) + 20*I*sqrt(2)*sqrt(b)*weierstrassPInverse(-4, 0, 
cos(f*x + e) + I*sin(f*x + e)) - 20*I*sqrt(2)*sqrt(b)*weierstrassPInverse( 
-4, 0, cos(f*x + e) - I*sin(f*x + e)))/f
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\text {Timed out} \] Input:

integrate((b*sec(f*x+e))**(1/2)*sin(f*x+e)**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{6} \,d x } \] Input:

integrate((b*sec(f*x+e))^(1/2)*sin(f*x+e)^6,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sec(f*x + e))*sin(f*x + e)^6, x)
 

Giac [F]

\[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{6} \,d x } \] Input:

integrate((b*sec(f*x+e))^(1/2)*sin(f*x+e)^6,x, algorithm="giac")
 

Output:

integrate(sqrt(b*sec(f*x + e))*sin(f*x + e)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^6\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \] Input:

int(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2),x)
 

Output:

int(sin(e + f*x)^6*(b/cos(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \sin \left (f x +e \right )^{6}d x \right ) \] Input:

int((b*sec(f*x+e))^(1/2)*sin(f*x+e)^6,x)
                                                                                    
                                                                                    
 

Output:

sqrt(b)*int(sqrt(sec(e + f*x))*sin(e + f*x)**6,x)