Integrand size = 27, antiderivative size = 75 \[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) f \sqrt {c+d \sin (e+f x)}} \] Output:
-2*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2)) *((c+d*sin(f*x+e))/(c+d))^(1/2)/(a+b)/f/(c+d*sin(f*x+e))^(1/2)
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) f \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]
Output:
(-2*EllipticPi[(2*b)/(a + b), (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[( c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])
Time = 0.46 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.
\(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {\sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}\) |
Input:
Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]
Output:
(2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*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. \(150\) vs. \(2(74)=148\).
Time = 0.77 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.01
method | result | size |
default | \(\frac {2 \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, -\frac {\left (c -d \right ) b}{a d -b c}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (a d -b c \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(151\) |
Input:
int(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(c-d)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d *(1+sin(f*x+e))/(c-d))^(1/2)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),-(c -d)*b/(a*d-b*c),((c-d)/(c+d))^(1/2))/(a*d-b*c)/cos(f*x+e)/(c+d*sin(f*x+e)) ^(1/2)/f
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\left (a + b \sin {\left (e + f x \right )}\right ) \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \] Input:
integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))**(1/2),x)
Output:
Integral(1/((a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x)
\[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)), x)
\[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)), x)
Timed out. \[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(1/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2)),x)
Output:
int(1/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2)), x)
\[ \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{2} b d +\sin \left (f x +e \right ) a d +\sin \left (f x +e \right ) b c +a c}d x \] Input:
int(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**2*b*d + sin(e + f*x)*a*d + sin (e + f*x)*b*c + a*c),x)