Integrand size = 32, antiderivative size = 107 \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b^2 f} \] Output:
2*A*(a-b)^(1/2)*(a+b)*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a-b)^(1 /2),((a-b)/(a+b))^(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e)) /(a-b))^(1/2)/b^2/f
Time = 7.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.97 \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {A (a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1+\sec (e+f x)}-\sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sqrt {\sec (e+f x)} \sin (e+f x)\right )}{b f \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sqrt {a+b \sec (e+f x)}} \] Input:
Integrate[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]
Output:
(A*(a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Sec[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*(Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Elli pticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 + Sec[e + f*x]] - Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[ e + f*x]))]*Sqrt[Sec[e + f*x]]*Sin[e + f*x]))/(b*f*((1 + Cos[e + f*x])^(-1 ))^(3/2)*Sqrt[a + b*Sec[e + f*x]])
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3042, 4492}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (A-A \csc \left (e+f x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right )}{b^2 f}\) |
Input:
Int[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]
Output:
(2*A*Sqrt[a - b]*(a + b)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a - b]], (a - b)/(a + b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]* Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f)
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(266\) vs. \(2(98)=196\).
Time = 26.76 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.50
| method | result | size |
| default | \(-\frac {2 A \left (\left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, a \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )+\left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )+\sin \left (f x +e \right ) \cos \left (f x +e \right ) a +b \sin \left (f x +e \right )\right ) \sqrt {a +b \sec \left (f x +e \right )}}{f b \left (\cos \left (f x +e \right )^{2} a +a \cos \left (f x +e \right )+b \cos \left (f x +e \right )+b \right )}\) | \(267\) |
| parts | \(-\frac {2 A \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}+\frac {2 A \left (\left (-\cos \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )+\left (-\cos \left (f x +e \right )^{2}-2 \cos \left (f x +e \right )-1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )+\left (\cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+1\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-\sin \left (f x +e \right ) \cos \left (f x +e \right ) a -b \sin \left (f x +e \right )\right ) \sqrt {a +b \sec \left (f x +e \right )}}{f b \left (\cos \left (f x +e \right )^{2} a +a \cos \left (f x +e \right )+b \cos \left (f x +e \right )+b \right )}\) | \(484\) |
Input:
int(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVER BOSE)
Output:
-2*A/f/b*((cos(f*x+e)^2+2*cos(f*x+e)+1)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)* (1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*a*EllipticE(cot(f*x+e)-csc (f*x+e),((a-b)/(a+b))^(1/2))+(cos(f*x+e)^2+2*cos(f*x+e)+1)*(cos(f*x+e)/(co s(f*x+e)+1))^(1/2)*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*b*Ellip ticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))+sin(f*x+e)*cos(f*x+e)*a+b* sin(f*x+e))*(a+b*sec(f*x+e))^(1/2)/(cos(f*x+e)^2*a+a*cos(f*x+e)+b*cos(f*x+ e)+b)
\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \] Input:
integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm= "fricas")
Output:
integral(-(A*sec(f*x + e)^2 - A*sec(f*x + e))/sqrt(b*sec(f*x + e) + a), x)
\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=- A \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx\right ) \] Input:
integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))**(1/2),x)
Output:
-A*(Integral(-sec(e + f*x)/sqrt(a + b*sec(e + f*x)), x) + Integral(sec(e + f*x)**2/sqrt(a + b*sec(e + f*x)), x))
\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \] Input:
integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm= "maxima")
Output:
-integrate((A*sec(f*x + e) - A)*sec(f*x + e)/sqrt(b*sec(f*x + e) + a), x)
\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \] Input:
integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm= "giac")
Output:
integrate(-(A*sec(f*x + e) - A)*sec(f*x + e)/sqrt(b*sec(f*x + e) + a), x)
Timed out. \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int \frac {A-\frac {A}{\cos \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int((A - A/cos(e + f*x))/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)),x)
Output:
int((A - A/cos(e + f*x))/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)), x)
\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=a \left (-\left (\int \frac {\sqrt {\sec \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{2}}{\sec \left (f x +e \right ) b +a}d x \right )+\int \frac {\sqrt {\sec \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right ) b +a}d x \right ) \] Input:
int(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x)
Output:
a*( - int((sqrt(sec(e + f*x)*b + a)*sec(e + f*x)**2)/(sec(e + f*x)*b + a), x) + int((sqrt(sec(e + f*x)*b + a)*sec(e + f*x))/(sec(e + f*x)*b + a),x))