Integrand size = 23, antiderivative size = 81 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=-\frac {2 b}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}}-\frac {2 E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \]
-2*b/f/(b*sec(f*x+e))^(3/2)/sin(f*x+e)^(1/2)+2*(sin(e+1/4*Pi+f*x)^2)^(1/2) /sin(e+1/4*Pi+f*x)*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 = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\frac {2 b \left (-1+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}\right )}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \]
(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 = 0.47 (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 |
\(\displaystyle -\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac {2 \sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\) |
(-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]])
3.5.64.3.1 Defintions of rubi rules used
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. \(355\) vs. \(2(96)=192\).
Time = 0.76 (sec) , antiderivative size = 356, normalized size of antiderivative = 4.40
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (1-\cos \left (f x +e \right )\right ) \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {2+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {2+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )-1\right ) \csc \left (f x +e \right )}{f \left (\frac {-\cot \left (f x +e \right )+\csc \left (f x +e \right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}\right )^{\frac {3}{2}} \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1\right ) \sqrt {-\frac {b \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )-1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )-1\right )}\) | \(356\) |
-1/f*2^(1/2)/(1/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(-cot(f*x+e)+csc(f*x+e)) )^(3/2)*(1-cos(f*x+e))/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(2*(-cot(f*x+e)+c sc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(cot(f*x+e)-csc(f*x +e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))-(-cot(f *x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(cot(f*x+e)- csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))+ (1-cos(f*x+e))^2*csc(f*x+e)^2-1)/(-b*((1-cos(f*x+e))^2*csc(f*x+e)^2+1)/((1 -cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*cs c(f*x+e)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=-\frac {2 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sqrt {\sin \left (f x + e\right )} + i \, \sqrt {i \, b} E(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) - i \, \sqrt {-i \, b} E(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) - i \, \sqrt {i \, b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + i \, \sqrt {-i \, b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right )}{b f \sin \left (f x + e\right )} \]
-(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))
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int \frac {1}{\sqrt {b \sec {\left (e + f x \right )}} \sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \]
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^{3/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]