Integrand size = 39, antiderivative size = 83 \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}} \] Output:
2*g*((d+c*cos(f*x+e))/(c+d))^(1/2)*EllipticPi(sin(1/2*f*x+1/2*e),2*a/(a+b) ,2^(1/2)*(c/(c+d))^(1/2))*(g*sec(f*x+e))^(1/2)/(a+b)/f/(c+d*sec(f*x+e))^(1 /2)
Time = 1.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}} \] Input:
Integrate[(g*Sec[e + f*x])^(3/2)/((a + b*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]
Output:
(2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (e + f*x )/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/((a + b)*f*Sqrt[c + d*Sec[e + f* x]])
Time = 0.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3042, 4463, 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.
\(\displaystyle \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (g \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4463 |
\(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {1}{(b+a \cos (e+f x)) \sqrt {d+c \cos (e+f x)}}dx}{\sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {1}{\left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{\sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {1}{(b+a \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}}dx}{\sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \int \frac {1}{\left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \sqrt {\frac {d}{c+d}+\frac {c \sin \left (e+f x+\frac {\pi }{2}\right )}{c+d}}}dx}{\sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)}}\) |
Input:
Int[(g*Sec[e + f*x])^(3/2)/((a + b*Sec[e + f*x])*Sqrt[c + d*Sec[e + f*x]]) ,x]
Output:
(2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (e + f*x )/2, (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/((a + b)*f*Sqrt[c + d*Sec[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]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[g*Sqr t[g*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]) Int [1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 5.78 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69
method | result | size |
default | \(\frac {2 i g \left (2 a \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -\frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right )-a \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right )-b \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right )\right ) \sqrt {g \sec \left (f x +e \right )}\, \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \cos \left (f x +e \right )}{f \left (a -b \right ) \left (a +b \right ) \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(223\) |
Input:
int((g*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x,method= _RETURNVERBOSE)
Output:
2*I*g/f/(a-b)/(a+b)*(2*a*EllipticPi(I*(cot(f*x+e)-csc(f*x+e)),-(a-b)/(a+b) ,I*((c-d)/(c+d))^(1/2))-a*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),(-(c-d)/(c+d ))^(1/2))-b*EllipticF(I*(cot(f*x+e)-csc(f*x+e)),(-(c-d)/(c+d))^(1/2)))*(g* sec(f*x+e))^(1/2)*(c+d*sec(f*x+e))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f* x+e)+1))^(1/2)*cos(f*x+e)/(d+c*cos(f*x+e))/(1/(cos(f*x+e)+1))^(1/2)
Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate((g*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \sec {\left (e + f x \right )}\right ) \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \] Input:
integrate((g*sec(f*x+e))**(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))**(1/2),x )
Output:
Integral((g*sec(e + f*x))**(3/2)/((a + b*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x)
\[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((g*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((g*sec(f*x + e))^(3/2)/((b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)), x)
\[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((g*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate((g*sec(f*x + e))^(3/2)/((b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)), x)
Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int((g/cos(e + f*x))^(3/2)/((a + b/cos(e + f*x))*(c + d/cos(e + f*x))^(1/2 )),x)
Output:
int((g/cos(e + f*x))^(3/2)/((a + b/cos(e + f*x))*(c + d/cos(e + f*x))^(1/2 )), x)
\[ \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\sqrt {g}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )}\, \sqrt {\sec \left (f x +e \right ) d +c}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right )^{2} b d +\sec \left (f x +e \right ) a d +\sec \left (f x +e \right ) b c +a c}d x \right ) g \] Input:
int((g*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))/(c+d*sec(f*x+e))^(1/2),x)
Output:
sqrt(g)*int((sqrt(sec(e + f*x))*sqrt(sec(e + f*x)*d + c)*sec(e + f*x))/(se c(e + f*x)**2*b*d + sec(e + f*x)*a*d + sec(e + f*x)*b*c + a*c),x)*g