Integrand size = 23, antiderivative size = 115 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=-\frac {2 b}{5 f (b \sec (e+f x))^{3/2} \sin ^{\frac {5}{2}}(e+f x)}-\frac {4 b}{5 f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}}-\frac {4 E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{5 f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \] Output:
-2/5*b/f/(b*sec(f*x+e))^(3/2)/sin(f*x+e)^(5/2)-4/5*b/f/(b*sec(f*x+e))^(3/2 )/sin(f*x+e)^(1/2)+4/5*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 = 0.85 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\frac {2 b \left (-2+\cos (2 (e+f x))+2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sin ^2(e+f x) \sqrt [4]{-\tan ^2(e+f x)}\right )}{5 f (b \sec (e+f x))^{3/2} \sin ^{\frac {5}{2}}(e+f x)} \] Input:
Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(7/2)),x]
Output:
(2*b*(-2 + Cos[2*(e + f*x)] + 2*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*Sin[e + f*x]^2*(-Tan[e + f*x]^2)^(1/4)))/(5*f*(b*Sec[e + f*x])^(3/ 2)*Sin[e + f*x]^(5/2))
Time = 0.98 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3064, 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.
| \(\displaystyle \int \frac {1}{\sin ^{\frac {7}{2}}(e+f x) \sqrt {b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^{7/2} \sqrt {b \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {3}{2}}(e+f x)}dx-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin (e+f x)^{3/2}}dx-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3064 |
| \(\displaystyle \frac {2}{5} \left (-2 \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (-2 \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}}dx-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle \frac {2}{5} \left (-\frac {2 \int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (-\frac {2 \int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)}dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {2}{5} \left (-\frac {2 \sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (-\frac {2 \sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{\sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2}{5} \left (-\frac {2 b}{f \sqrt {\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac {2 \sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}}\right )-\frac {2 b}{5 f \sin ^{\frac {5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}\) |
Input:
Int[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(7/2)),x]
Output:
(-2*b)/(5*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(5/2)) + (2*((-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]])))/5
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. \(247\) vs. \(2(96)=192\).
Time = 0.32 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(-\frac {2 \left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-2 \sin \left (f x +e \right )^{2}-2 \tan \left (f x +e \right ) \sin \left (f x +e \right )\right )+\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\sin \left (f x +e \right )^{2}+\tan \left (f x +e \right ) \sin \left (f x +e \right )\right )+2 \sin \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{5 f \sqrt {b \sec \left (f x +e \right )}\, \sin \left (f x +e \right )^{\frac {5}{2}}}\) | \(248\) |
Input:
int(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/5/f/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(5/2)*((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)*El lipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*(-2*sin(f*x+e)^2-2*ta n(f*x+e)*sin(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))*(sin(f*x+e)^2+tan(f*x+e)*sin(f*x+e))+2*sin(f*x +e)^2+cos(f*x+e))
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.08 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=-\frac {2 \, {\left ({\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {i \, b} E(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {-i \, b} E(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {i \, b} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {-i \, b} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) \sin \left (f x + e\right ) + {\left (2 \, \cos \left (f x + e\right )^{4} - 3 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}\right )}}{5 \, {\left (b f \cos \left (f x + e\right )^{2} - b f\right )} \sin \left (f x + e\right )} \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(7/2),x, algorithm="fricas")
Output:
-2/5*((I*cos(f*x + e)^2 - I)*sqrt(I*b)*elliptic_e(arcsin(cos(f*x + e) + I* sin(f*x + e)), -1)*sin(f*x + e) + (-I*cos(f*x + e)^2 + I)*sqrt(-I*b)*ellip tic_e(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1)*sin(f*x + e) + (-I*cos(f* x + e)^2 + I)*sqrt(I*b)*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1)*sin(f*x + e) + (I*cos(f*x + e)^2 - I)*sqrt(-I*b)*elliptic_f(arcsin(cos (f*x + e) - I*sin(f*x + e)), -1)*sin(f*x + e) + (2*cos(f*x + e)^4 - 3*cos( f*x + e)^2)*sqrt(b/cos(f*x + e))*sqrt(sin(f*x + e)))/((b*f*cos(f*x + e)^2 - b*f)*sin(f*x + e))
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\text {Timed out} \] Input:
integrate(1/(b*sec(f*x+e))**(1/2)/sin(f*x+e)**(7/2),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(7/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*sec(f*x + e))*sin(f*x + e)^(7/2)), x)
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\text {Timed out} \] Input:
integrate(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(7/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^{7/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(1/(sin(e + f*x)^(7/2)*(b/cos(e + f*x))^(1/2)),x)
Output:
int(1/(sin(e + f*x)^(7/2)*(b/cos(e + f*x))^(1/2)), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {7}{2}}(e+f x)} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right ) \sin \left (f x +e \right )^{4}}d x \right )}{b} \] Input:
int(1/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(7/2),x)
Output:
(sqrt(b)*int((sqrt(sin(e + f*x))*sqrt(sec(e + f*x)))/(sec(e + f*x)*sin(e + f*x)**4),x))/b