\(\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx\) [456]

Optimal result

Integrand size = 25, antiderivative size = 128 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=-\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 a^4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {a \sin (e+f x)}} \] Output:

-5/6*a^3*b*(a*sin(f*x+e))^(1/2)/f/(b*sec(f*x+e))^(1/2)-1/3*a*b*(a*sin(f*x+ 
e))^(5/2)/f/(b*sec(f*x+e))^(1/2)+5/12*a^4*InverseJacobiAM(e-1/4*Pi+f*x,2^( 
1/2))*(b*sec(f*x+e))^(1/2)*sin(2*f*x+2*e)^(1/2)/f/(a*sin(f*x+e))^(1/2)
 

Mathematica [C] (verified)

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

Time = 17.78 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\frac {a^3 b \sqrt {a \sin (e+f x)} \left (2 (-6+\cos (2 (e+f x)))+5 \csc ^2(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{12 f \sqrt {b \sec (e+f x)}} \] Input:

Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2),x]
 

Output:

(a^3*b*Sqrt[a*Sin[e + f*x]]*(2*(-6 + Cos[2*(e + f*x)]) + 5*Csc[e + f*x]^2* 
Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4))) 
/(12*f*Sqrt[b*Sec[e + f*x]])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3063, 3042, 3063, 3042, 3065, 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.

\(\displaystyle \int (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}dx\)

\(\Big \downarrow \) 3063

\(\displaystyle \frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3063

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3065

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

Int[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2),x]
 

Output:

-1/3*(a*b*(a*Sin[e + f*x])^(5/2))/(f*Sqrt[b*Sec[e + f*x]]) + (5*a^2*(-((a* 
b*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])) + (a^2*EllipticF[e - Pi/ 
4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[a*Sin[e 
 + f*x]])))/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 3063

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

rule 3065

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n   Int[(a*Sin[e 
+ f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && Int 
egerQ[m - 1/2] && IntegerQ[n - 1/2]
 

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] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.84 (sec) , antiderivative size = 990, normalized size of antiderivative = 7.73

method	result	size
default	\(\text {Expression too large to display}\)	\(990\)

Input:

int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/384/f*a^3*(sin(f*x+e)*cos(f*x+e)*(16*cos(f*x+e)^2-56)+(-6*cos(f*x+e)-6)* 
(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc 
(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1 
/2*I,1/2*2^(1/2))+(-6*cos(f*x+e)-6)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*cs 
c(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi(( 
csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+(32*cos(f*x+e)+32)*( 
csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(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))+I*(-6*cos(f*x+e)-6)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2 
*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((csc(f*x+e) 
-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+I*(6*cos(f*x+e)+6)*(csc(f*x+e) 
-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot 
(f*x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^ 
(1/2))+(3*cos(f*x+e)+3)*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)* 
ln(-(cos(f*x+e)*cot(f*x+e)-2*cot(f*x+e)+2*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f 
*x+e)+1)^2)^(1/2)*sin(f*x+e)+csc(f*x+e)-2*cos(f*x+e)-sin(f*x+e)+2)/(-1+cos 
(f*x+e)))+(-3*cos(f*x+e)-3)*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1 
/2)*ln(-(cos(f*x+e)*cot(f*x+e)-2*cot(f*x+e)-2*(-2*sin(f*x+e)*cos(f*x+e)/(c 
os(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+csc(f*x+e)-2*cos(f*x+e)-sin(f*x+e)+2)/(-1 
+cos(f*x+e)))+(-6*cos(f*x+e)-6)*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1...
 

Fricas [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \] Input:

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(7/2),x, algorithm="fricas")
 

Output:

integral(-(a^3*cos(f*x + e)^2 - a^3)*sqrt(b*sec(f*x + e))*sqrt(a*sin(f*x + 
 e))*sin(f*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate((b*sec(f*x+e))**(1/2)*(a*sin(f*x+e))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \] Input:

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(7/2), x)
 

Giac [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \] Input:

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \] Input:

int((a*sin(e + f*x))^(7/2)*(b/cos(e + f*x))^(1/2),x)
 

Output:

int((a*sin(e + f*x))^(7/2)*(b/cos(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\sqrt {b}\, \sqrt {a}\, \left (\int \sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}\, \sin \left (f x +e \right )^{3}d x \right ) a^{3} \] Input:

int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(7/2),x)
 

Output:

sqrt(b)*sqrt(a)*int(sqrt(sin(e + f*x))*sqrt(sec(e + f*x))*sin(e + f*x)**3, 
x)*a**3