Integrand size = 25, antiderivative size = 130 \[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=-\frac {2 b}{7 a f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}-\frac {4 b}{7 a^3 f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}+\frac {4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{7 a^4 f \sqrt {a \sin (e+f x)}} \] Output:
-2/7*b/a/f/(b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(7/2)-4/7*b/a^3/f/(b*sec(f* x+e))^(1/2)/(a*sin(f*x+e))^(3/2)+4/7*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)/a^4/f/(a*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=-\frac {2 \cos (2 (e+f x)) (b \sec (e+f x))^{3/2} \left ((-2+\cos (2 (e+f x))) \csc ^2(e+f x)+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{7 a^3 b f \left (-2+\sec ^2(e+f x)\right ) (a \sin (e+f x))^{3/2}} \] Input:
Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(9/2),x]
Output:
(-2*Cos[2*(e + f*x)]*(b*Sec[e + f*x])^(3/2)*((-2 + Cos[2*(e + f*x)])*Csc[e + f*x]^2 + 2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f *x]^2)^(3/4)))/(7*a^3*b*f*(-2 + Sec[e + f*x]^2)*(a*Sin[e + f*x])^(3/2))
Time = 1.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3064, 3042, 3064, 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.
| \(\displaystyle \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle \frac {6 \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}}dx}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}}dx}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle \frac {6 \left (\frac {2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (\frac {2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 a^2 \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 a^2 \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{3 a^2 f \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}\right )}{7 a^2}-\frac {2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(9/2),x]
Output:
(-2*b)/(7*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2)) + (6*((-2*b)/(3 *a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2)) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^2*f*Sqrt[a*Sin [e + f*x]])))/(7*a^2)
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]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1)/ (a*f*(m + 1))), x] + Simp[(m - n + 2)/(a^2*(m + 1)) Int[(a*Sin[e + f*x])^ (m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n]
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]
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]
Time = 0.61 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {2 \sqrt {b \sec \left (f x +e \right )}\, \left (\left (-2 \cos \left (f x +e \right )-2\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cot \left (f x +e \right )^{3}+3 \cot \left (f x +e \right ) \csc \left (f x +e \right )^{2}\right )}{7 f \sqrt {a \sin \left (f x +e \right )}\, a^{4}}\) | \(146\) |
Input:
int((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
Output:
-2/7/f*(b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(1/2)/a^4*((-2*cos(f*x+e)-2)*(c sc(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))-2*cot(f*x+e)^3+3*cot(f*x+e)*csc(f*x+e)^2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + 2 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - {\left (2 \, \cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{7 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} - 2 \, a^{5} f \cos \left (f x + e\right )^{2} + a^{5} f\right )}} \] Input:
integrate((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x, algorithm="fricas")
Output:
-2/7*(2*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(I*a*b)*elliptic_f(arc sin(cos(f*x + e) + I*sin(f*x + e)), -1) + 2*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-I*a*b)*elliptic_f(arcsin(cos(f*x + e) - I*sin(f*x + e)), - 1) - (2*cos(f*x + e)^3 - 3*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f *x + e)))/(a^5*f*cos(f*x + e)^4 - 2*a^5*f*cos(f*x + e)^2 + a^5*f)
Timed out. \[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((b*sec(f*x+e))**(1/2)/(a*sin(f*x+e))**(9/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*sec(f*x + e))/(a*sin(f*x + e))^(9/2), x)
\[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x, algorithm="giac")
Output:
integrate(sqrt(b*sec(f*x + e))/(a*sin(f*x + e))^(9/2), x)
Timed out. \[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:
int((b/cos(e + f*x))^(1/2)/(a*sin(e + f*x))^(9/2),x)
Output:
int((b/cos(e + f*x))^(1/2)/(a*sin(e + f*x))^(9/2), x)
\[ \int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sin \left (f x +e \right )^{5}}d x \right )}{a^{5}} \] Input:
int((b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(9/2),x)
Output:
(sqrt(b)*sqrt(a)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/sin(e + f*x)* *5,x))/a**5