Integrand size = 21, antiderivative size = 62 \[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}+\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \] Output:
-b*csc(f*x+e)/f/(b*sec(f*x+e))^(1/2)+cos(f*x+e)^(1/2)*InverseJacobiAM(1/2* f*x+1/2*e,2^(1/2))*(b*sec(f*x+e))^(1/2)/f
Time = 0.46 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (-\cot (e+f x)+\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {b \sec (e+f x)}}{f} \] Input:
Integrate[Csc[e + f*x]^2*Sqrt[b*Sec[e + f*x]],x]
Output:
((-Cot[e + f*x] + Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[b*Sec [e + f*x]])/f
Time = 0.57 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3105, 3042, 4258, 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.
\(\displaystyle \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^2 \sqrt {b \sec (e+f x)}dx\) |
\(\Big \downarrow \) 3105 |
\(\displaystyle \frac {1}{2} \int \sqrt {b \sec (e+f x)}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f}-\frac {b \csc (e+f x)}{f \sqrt {b \sec (e+f x)}}\) |
Input:
Int[Csc[e + f*x]^2*Sqrt[b*Sec[e + f*x]],x]
Output:
-((b*Csc[e + f*x])/(f*Sqrt[b*Sec[e + f*x]])) + (Sqrt[Cos[e + f*x]]*Ellipti cF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f
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*(m - 1))), x] + Simp[a^2*((m + n - 2)/(m - 1)) Int[(a*Csc[e + f* x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[ m, 1] && IntegersQ[2*m, 2*n] && !GtQ[n, m]
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]
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]
Result contains complex when optimal does not.
Time = 3.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {\sqrt {b \sec \left (f x +e \right )}\, \left (i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )-\cot \left (f x +e \right )\right )}{f}\) | \(87\) |
Input:
int(csc(f*x+e)^2*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/f*(b*sec(f*x+e))^(1/2)*(I*(cos(f*x+e)+1)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f *x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),I)-cot(f*x +e))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.58 \[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {-i \, \sqrt {2} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, f \sin \left (f x + e\right )} \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
1/2*(-I*sqrt(2)*sqrt(b)*sin(f*x + e)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(2)*sqrt(b)*sin(f*x + e)*weierstrassPInverse( -4, 0, cos(f*x + e) - I*sin(f*x + e)) - 2*sqrt(b/cos(f*x + e))*cos(f*x + e ))/(f*sin(f*x + e))
\[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate(csc(f*x+e)**2*(b*sec(f*x+e))**(1/2),x)
Output:
Integral(sqrt(b*sec(e + f*x))*csc(e + f*x)**2, x)
\[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^2, x)
\[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^2, x)
Timed out. \[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \] Input:
int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^2,x)
Output:
int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^2, x)
\[ \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx=\sqrt {b}\, \left (\int \sqrt {\sec \left (f x +e \right )}\, \csc \left (f x +e \right )^{2}d x \right ) \] Input:
int(csc(f*x+e)^2*(b*sec(f*x+e))^(1/2),x)
Output:
sqrt(b)*int(sqrt(sec(e + f*x))*csc(e + f*x)**2,x)