Integrand size = 23, antiderivative size = 81 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=-\frac {2 b}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}}-\frac {2 E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \] Output:
-2*b/f/(b*sec(f*x+e))^(3/2)/sin(f*x+e)^(1/2)+2*EllipticE(cos(e+1/4*Pi+f*x) ,2^(1/2))*sin(f*x+e)^(1/2)/f/(b*sec(f*x+e))^(1/2)/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.58 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {2 b \left (-1+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}\right )}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \] Input:
Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(3/2)),x]
Output:
(2*b*(-1 + Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x ]^2)^(1/4)))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])
Time = 0.77 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3064, 3042, 3065, 3042, 3052, 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.
| \(\displaystyle \int \frac {1}{\sin ^{\frac {3}{2}}(e+f x) \sqrt {b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^{3/2} \sqrt {b \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle -2 \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle -\frac {2 \int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {2 \sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac {2 \sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(3/2)),x]
Output:
(-2*b)/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]) - (2*EllipticE[e - Pi /4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2* f*x]])
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(72)=144\).
Time = 0.24 (sec) , antiderivative size = 361, normalized size of antiderivative = 4.46
| method | result | size |
| default | \(\frac {2 \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )-\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )-2}{f \sqrt {b \sec \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}}\) | \(361\) |
Input:
int(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/f/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(1/2)*(2*(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)*Ell ipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))-(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))+2*(csc(f*x+e)-c ot(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)*EllipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*sec(f* x+e)-(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))*sec(f*x+e)-2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=-\frac {2 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sqrt {\sin \left (f x + e\right )} + i \, \sqrt {i \, b} E(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) - i \, \sqrt {-i \, b} E(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) - i \, \sqrt {i \, b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + i \, \sqrt {-i \, b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right )}{b f \sin \left (f x + e\right )} \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(3/2),x, algorithm="fricas")
Output:
-(2*sqrt(b/cos(f*x + e))*cos(f*x + e)^2*sqrt(sin(f*x + e)) + I*sqrt(I*b)*e lliptic_e(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1)*sin(f*x + e) - I*sqrt (-I*b)*elliptic_e(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1)*sin(f*x + e) - I*sqrt(I*b)*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1)*sin(f* x + e) + I*sqrt(-I*b)*elliptic_f(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1 )*sin(f*x + e))/(b*f*sin(f*x + e))
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int \frac {1}{\sqrt {b \sec {\left (e + f x \right )}} \sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \] Input:
integrate(1/(b*sec(f*x+e))**(1/2)/sin(f*x+e)**(3/2),x)
Output:
Integral(1/(sqrt(b*sec(e + f*x))*sin(e + f*x)**(3/2)), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(3/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*sec(f*x + e))*sin(f*x + e)^(3/2)), x)
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\text {Timed out} \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^{3/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(1/(sin(e + f*x)^(3/2)*(b/cos(e + f*x))^(1/2)),x)
Output:
int(1/(sin(e + f*x)^(3/2)*(b/cos(e + f*x))^(1/2)), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right ) \sin \left (f x +e \right )^{2}}d x \right )}{b} \] Input:
int(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(3/2),x)
Output:
(sqrt(b)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/(sec(e + f*x)*sin(e + f*x)**2),x))/b