Integrand size = 25, antiderivative size = 137 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=-\frac {2}{7 a b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}}+\frac {2}{21 a^3 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}}-\frac {2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{21 a^4 b^2 f \sqrt {a \sin (e+f x)}} \] Output:
-2/7/a/b/f/(b*sec(f*x+e))^(1/2)/(a*sin(f*x+e))^(7/2)+2/21/a^3/b/f/(b*sec(f *x+e))^(1/2)/(a*sin(f*x+e))^(3/2)-2/21*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/b^2/f/(a*sin(f*x+e))^(1/2 )
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.93 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\frac {\cos (2 (e+f x)) \csc ^4(e+f x) \sqrt {a \sin (e+f x)} \left ((5+\cos (2 (e+f x))) \sec ^2(e+f x)-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{7/4}\right )}{21 a^5 b f \sqrt {b \sec (e+f x)} \left (-2+\sec ^2(e+f x)\right )} \] Input:
Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(9/2)),x]
Output:
(Cos[2*(e + f*x)]*Csc[e + f*x]^4*Sqrt[a*Sin[e + f*x]]*((5 + Cos[2*(e + f*x )])*Sec[e + f*x]^2 - 2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(- Tan[e + f*x]^2)^(7/4)))/(21*a^5*b*f*Sqrt[b*Sec[e + f*x]]*(-2 + Sec[e + f*x ]^2))
Time = 1.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3061, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x))^{9/2} (b \sec (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x))^{9/2} (b \sec (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3061 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}}dx}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}}dx}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle -\frac {\frac {2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle -\frac {\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx}{3 a^2}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 a^2 \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 a^2 \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{3 a^2 f \sqrt {a \sin (e+f x)}}-\frac {2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt {b \sec (e+f x)}}}{7 a^2 b^2}-\frac {2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(9/2)),x]
Output:
-2/(7*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2)) - ((-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*b^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[(a*Sin[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n + 1)/(a *b*f*(m + 1))), x] - Simp[(n + 1)/(a^2*b^2*(m + 1)) Int[(a*Sin[e + f*x])^ (m + 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n , -1] && 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*(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 = 0.96 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\frac {2 \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-1-\sec \left (f x +e \right )\right )}{21}+\frac {2 \left (-\cos \left (f x +e \right )^{2}-2\right ) \csc \left (f x +e \right )^{3}}{21}}{f \sqrt {a \sin \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}\, a^{4} b}\) | \(144\) |
Input:
int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
Output:
2/21/f/(a*sin(f*x+e))^(1/2)/(b*sec(f*x+e))^(1/2)/a^4/b*((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))*(-1-sec(f*x +e))+(-cos(f*x+e)^2-2)*csc(f*x+e)^3)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\frac {2 \, {\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-i \, a b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - {\left (\cos \left (f x + e\right )^{3} + 2 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{21 \, {\left (a^{5} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{5} b^{2} f \cos \left (f x + e\right )^{2} + a^{5} b^{2} f\right )}} \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="fricas ")
Output:
2/21*((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(I*a*b)*elliptic_f(arcsi n(cos(f*x + e) + I*sin(f*x + e)), -1) + (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) - (cos(f*x + e)^3 + 2*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e )))/(a^5*b^2*f*cos(f*x + e)^4 - 2*a^5*b^2*f*cos(f*x + e)^2 + a^5*b^2*f)
Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*sec(f*x+e))**(3/2)/(a*sin(f*x+e))**(9/2),x)
Output:
Timed out
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="maxima ")
Output:
integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(9/2)), x)
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\int { \frac {1}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x, algorithm="giac")
Output:
integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(9/2)), x)
Timed out. \[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\int \frac {1}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(1/((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(3/2)),x)
Output:
int(1/((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(3/2)), x)
\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )^{2} \sin \left (f x +e \right )^{5}}d x \right )}{a^{5} b^{2}} \] Input:
int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(9/2),x)
Output:
(sqrt(b)*sqrt(a)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/(sec(e + f*x) **2*sin(e + f*x)**5),x))/(a**5*b**2)