\(\int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx\) [393]

Optimal result

Integrand size = 32, antiderivative size = 107 \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b^2 f} \]

[Out]

2*A*(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))*(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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {4089} \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right )}{b^2 f} \]

[In]

Int[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(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)

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(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 + C
sc[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]

Rubi steps \begin{align*} \text {integral}& = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.92 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.97 \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {A (a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1+\sec (e+f x)}-\sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sqrt {\sec (e+f x)} \sin (e+f x)\right )}{b f \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sqrt {a+b \sec (e+f x)}} \]

[In]

Integrate[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(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])]*EllipticE[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]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(98)=196\).

Time = 21.68 (sec) , antiderivative size = 565, normalized size of antiderivative = 5.28

method	result	size
default	\(-\frac {2 A \left (\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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a \cos \left (f x +e \right )^{2}+\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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b \cos \left (f x +e \right )^{2}+2 \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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a \cos \left (f x +e \right )+2 \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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b \cos \left (f x +e \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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a +\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}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b +\cos \left (f x +e \right ) \sin \left (f x +e \right ) a +\sin \left (f x +e \right ) b \right ) \sqrt {a +b \sec \left (f x +e \right )}}{f b \left (b +a \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )}\)	\(565\)
parts	\(\text {Expression too large to display}\)	\(935\)

[In]

int(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*A/f/b*((1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+
e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a*cos(f*x+e)^2+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b*cos(f*x+e)^2+2*(1/(a+b)*(b+a*cos(
f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^
(1/2))*a*cos(f*x+e)+2*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ellipt
icE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b*cos(f*x+e)+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c
os(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a+(1/(a+b)*(b+a*cos(f*x+e
))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2)
)*b+cos(f*x+e)*sin(f*x+e)*a+sin(f*x+e)*b)*(a+b*sec(f*x+e))^(1/2)/(b+a*cos(f*x+e))/(cos(f*x+e)+1)

Fricas [F]

\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(A*sec(f*x + e)^2 - A*sec(f*x + e))/sqrt(b*sec(f*x + e) + a), x)

Sympy [F]

\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=- A \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx\right ) \]

[In]

integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))**(1/2),x)

[Out]

-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
))

Maxima [F]

\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

-integrate((A*sec(f*x + e) - A)*sec(f*x + e)/sqrt(b*sec(f*x + e) + a), x)

Giac [F]

\[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int { -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate(sec(f*x+e)*(A-A*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(-(A*sec(f*x + e) - A)*sec(f*x + e)/sqrt(b*sec(f*x + e) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx=\int \frac {A-\frac {A}{\cos \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int((A - A/cos(e + f*x))/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)),x)

[Out]

int((A - A/cos(e + f*x))/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)), x)