Integrand size = 21, antiderivative size = 63 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \] Output:
-2*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*s in(f*x+e))/(a+b))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)
Time = 0.77 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \] Input:
Integrate[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]
Output:
(-2*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3286, 3042, 3284}
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.
\(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {\csc (e+f x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{\sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sin (e+f x) \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{\sqrt {a+b \sin (e+f x)}}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}\) |
Input:
Int[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]
Output:
(2*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f* x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(62)=124\).
Time = 0.37 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.14
method | result | size |
default | \(-\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {b \left (-1+\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{a \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\) | \(135\) |
Input:
int(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-b*(-1+sin(f*x+e))/(a+b))^(1/2)*( -b*(1+sin(f*x+e))/(a-b))^(1/2)*EllipticPi(((a+b*sin(f*x+e))/(a-b))^(1/2),( a-b)/a,((a-b)/(a+b))^(1/2))/a/cos(f*x+e)/(a+b*sin(f*x+e))^(1/2)/f
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))**(1/2),x)
Output:
Integral(csc(e + f*x)/sqrt(a + b*sin(e + f*x)), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(csc(f*x + e)/sqrt(b*sin(f*x + e) + a), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \] Input:
integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(csc(f*x + e)/sqrt(b*sin(f*x + e) + a), x)
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)),x)
Output:
int(1/(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2)), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) b +a}\, \csc \left (f x +e \right )}{\sin \left (f x +e \right ) b +a}d x \] Input:
int(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x)
Output:
int((sqrt(sin(e + f*x)*b + a)*csc(e + f*x))/(sin(e + f*x)*b + a),x)