Integrand size = 19, antiderivative size = 62 \[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \] Output:
sec(b*x+a)/b/csc(b*x+a)^(3/2)+csc(b*x+a)^(1/2)*EllipticE(cos(1/2*a+1/4*Pi+ 1/2*b*x),2^(1/2))*sin(b*x+a)^(1/2)/b
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {\sqrt {\csc (a+b x)} \left (E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}+\sin (a+b x) \tan (a+b x)\right )}{b} \] Input:
Integrate[Sec[a + b*x]^2/Sqrt[Csc[a + b*x]],x]
Output:
(Sqrt[Csc[a + b*x]]*(EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x] ] + Sin[a + b*x]*Tan[a + b*x]))/b
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 3106, 3042, 4258, 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.
\(\displaystyle \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (a+b x)^2}{\sqrt {\csc (a+b x)}}dx\) |
\(\Big \downarrow \) 3106 |
\(\displaystyle \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \int \frac {1}{\sqrt {\csc (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \int \frac {1}{\sqrt {\csc (a+b x)}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {1}{2} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sec (a+b x)}{b \csc ^{\frac {3}{2}}(a+b x)}-\frac {\sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b}\) |
Input:
Int[Sec[a + b*x]^2/Sqrt[Csc[a + b*x]],x]
Output:
Sec[a + b*x]/(b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*EllipticE[(a - P i/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/b
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(n - 1))), x] + Simp[b^2*((m + n - 2)/(n - 1)) Int[(a*Csc[e + f*x]) ^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(55)=110\).
Time = 0.76 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.85
method | result | size |
default | \(\frac {\sqrt {\cos \left (b x +a \right )^{2} \sin \left (b x +a \right )}\, \left (2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (b x +a \right )^{2}+2\right )}{2 \sqrt {-\sin \left (b x +a \right ) \left (\sin \left (b x +a \right )-1\right ) \left (\sin \left (b x +a \right )+1\right )}\, \cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(177\) |
Input:
int(sec(b*x+a)^2/csc(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(cos(b*x+a)^2*sin(b*x+a))^(1/2)*(2*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a) +2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))- (sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF ((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))-2*cos(b*x+a)^2+2)/(-sin(b*x+a)*(sin(b*x +a)-1)*(sin(b*x+a)+1))^(1/2)/cos(b*x+a)/sin(b*x+a)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55 \[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=-\frac {\sqrt {2 i} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )}}{\sqrt {\sin \left (b x + a\right )}}}{2 \, b \cos \left (b x + a\right )} \] Input:
integrate(sec(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="fricas")
Output:
-1/2*(sqrt(2*I)*cos(b*x + a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) + sqrt(-2*I)*cos(b*x + a)*weierstrassZe ta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a))) + 2*(co s(b*x + a)^2 - 1)/sqrt(sin(b*x + a)))/(b*cos(b*x + a))
\[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\sec ^{2}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \] Input:
integrate(sec(b*x+a)**2/csc(b*x+a)**(1/2),x)
Output:
Integral(sec(a + b*x)**2/sqrt(csc(a + b*x)), x)
\[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\sec \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(sec(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sec(b*x + a)^2/sqrt(csc(b*x + a)), x)
\[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\sec \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(sec(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(sec(b*x + a)^2/sqrt(csc(b*x + a)), x)
Timed out. \[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {1}{{\cos \left (a+b\,x\right )}^2\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \] Input:
int(1/(cos(a + b*x)^2*(1/sin(a + b*x))^(1/2)),x)
Output:
int(1/(cos(a + b*x)^2*(1/sin(a + b*x))^(1/2)), x)
\[ \int \frac {\sec ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\sqrt {\csc \left (b x +a \right )}\, \sec \left (b x +a \right )^{2}}{\csc \left (b x +a \right )}d x \] Input:
int(sec(b*x+a)^2/csc(b*x+a)^(1/2),x)
Output:
int((sqrt(csc(a + b*x))*sec(a + b*x)**2)/csc(a + b*x),x)