\(\int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx\) [391]

Optimal result

Integrand size = 21, antiderivative size = 128 \[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=-\frac {16 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac {20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)} \sin ^5(e+f x)}{f} \] Output:

-16/3*b^2*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec( 
f*x+e))^(1/2)+8/3*b^3*sin(f*x+e)/f/(b*sec(f*x+e))^(3/2)+20/9*b^3*sin(f*x+e 
)^3/f/(b*sec(f*x+e))^(3/2)+2*b*(b*sec(f*x+e))^(1/2)*sin(f*x+e)^5/f
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=-\frac {b \sqrt {b \sec (e+f x)} \left (384 \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )-158 \sin (e+f x)-13 \sin (3 (e+f x))+\sin (5 (e+f x))\right )}{72 f} \] Input:

Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6,x]
 

Output:

-1/72*(b*Sqrt[b*Sec[e + f*x]]*(384*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/ 
2, 2] - 158*Sin[e + f*x] - 13*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/f
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3104, 3042, 3107, 3042, 3107, 3042, 4258, 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.

\(\displaystyle \int \sin ^6(e+f x) (b \sec (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(b \sec (e+f x))^{3/2}}{\csc (e+f x)^6}dx\)

\(\Big \downarrow \) 3104

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \int \frac {1}{\csc (e+f x)^4 \sqrt {b \sec (e+f x)}}dx\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \int \frac {1}{\csc (e+f x)^2 \sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3107

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \left (\frac {2 \int \sqrt {\cos (e+f x)}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \left (\frac {2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 b \sin ^5(e+f x) \sqrt {b \sec (e+f x)}}{f}-10 b^2 \left (\frac {2}{3} \left (\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}\right )-\frac {2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\right )\)

Input:

Int[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6,x]
 

Output:

(2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^5)/f - 10*b^2*((-2*b*Sin[e + f*x]^3 
)/(9*f*(b*Sec[e + f*x])^(3/2)) + (2*((4*EllipticE[(e + f*x)/2, 2])/(5*f*Sq 
rt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x])/(5*f*(b*Sec[e 
+ f*x])^(3/2))))/3)
 

Defintions of rubi rules used

rule 3042

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

rule 3104

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)/ 
(f*a*(n - 1))), x] + Simp[b^2*((m + 1)/(a^2*(n - 1)))   Int[(a*Csc[e + f*x] 
)^(m + 2)*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ 
[n, 1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
 

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 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]
 

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 = 7.04 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.79

method	result	size
default	\(-\frac {2 b \left (\left (\cos \left (f x +e \right )^{5}+\cos \left (f x +e \right )^{4}-4 \cos \left (f x +e \right )^{3}-4 \cos \left (f x +e \right )^{2}+15 \cos \left (f x +e \right )-9\right ) \sin \left (f x +e \right )+24 i \left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )+24 i \left (-\cos \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )\right ) \sqrt {b \sec \left (f x +e \right )}}{9 f \left (\cos \left (f x +e \right )+1\right )}\)	\(229\)

Input:

int((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x,method=_RETURNVERBOSE)
 

Output:

-2/9/f*b*((cos(f*x+e)^5+cos(f*x+e)^4-4*cos(f*x+e)^3-4*cos(f*x+e)^2+15*cos( 
f*x+e)-9)*sin(f*x+e)+24*I*(cos(f*x+e)^2+2*cos(f*x+e)+1)*(1/(cos(f*x+e)+1)) 
^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e 
)),I)+24*I*(-cos(f*x+e)^2-2*cos(f*x+e)-1)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2 
)*(1/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(cot(f*x+e)-csc(f*x+e)),I))*(b*sec( 
f*x+e))^(1/2)/(cos(f*x+e)+1)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=-\frac {2 \, {\left (12 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 12 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (b \cos \left (f x + e\right )^{4} - 4 \, b \cos \left (f x + e\right )^{2} - 9 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{9 \, f} \] Input:

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="fricas")
 

Output:

-2/9*(12*I*sqrt(2)*b^(3/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 
0, cos(f*x + e) + I*sin(f*x + e))) - 12*I*sqrt(2)*b^(3/2)*weierstrassZeta( 
-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (b*cos 
(f*x + e)^4 - 4*b*cos(f*x + e)^2 - 9*b)*sqrt(b/cos(f*x + e))*sin(f*x + e)) 
/f
 

Sympy [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=\text {Timed out} \] Input:

integrate((b*sec(f*x+e))**(3/2)*sin(f*x+e)**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{6} \,d x } \] Input:

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="maxima")
 

Output:

integrate((b*sec(f*x + e))^(3/2)*sin(f*x + e)^6, x)
 

Giac [F]

\[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{6} \,d x } \] Input:

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="giac")
 

Output:

integrate((b*sec(f*x + e))^(3/2)*sin(f*x + e)^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^6\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \] Input:

int(sin(e + f*x)^6*(b/cos(e + f*x))^(3/2),x)
 

Output:

int(sin(e + f*x)^6*(b/cos(e + f*x))^(3/2), x)
 

Reduce [F]

\[ \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \sec \left (f x +e \right ) \sin \left (f x +e \right )^{6}d x \right ) b \] Input:

int((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x)
 

Output:

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