3.4.92 \(\int \frac {\sec (e+f x) (A+A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx\) [392]

Optimal. Leaf size=105 \[ -\frac {2 A (a-b) \sqrt {a+b} \cot (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b^2 f} \]

[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

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Rubi [A]
time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {4089} \begin {gather*} -\frac {2 A (a-b) \sqrt {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 (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(a - b)*Sqrt[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*} \int \frac {\sec (e+f x) (A+A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx &=-\frac {2 A (a-b) \sqrt {a+b} \cot (e+f x) E\left (\sin ^{-1}\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(105)=210\).
time = 9.85, size = 248, normalized size = 2.36 \begin {gather*} \frac {A (1+\sec (e+f x)) \left (2 (b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)} \left (\frac {\sqrt {\frac {a-b}{a+b}} (a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} E\left (\text {ArcSin}\left (\sqrt {\frac {a-b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a+b}{a-b}\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}+(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {\sec (e+f x)}}\right )}{b f \sqrt {a+b \sec (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(A*(1 + Sec[e + f*x])*(2*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2] + (Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/
2]^2*Sec[e + f*x]]*((Sqrt[(a - b)/(a + b)]*(a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Ell
ipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]
+ (b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2))/Sqrt[Sec[e + f*x]]))/(b*f*Sqrt[a + b*Sec[
e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(96)=192\).
time = 11.81, size = 642, normalized size = 6.11

method	result	size
default	\(-\frac {2 A \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (2 \cos \left (f x +e \right ) \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) b -\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, a -\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, b +2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b \sin \left (f x +e \right )-\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, a -\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, b +\left (\cos ^{2}\left (f x +e \right )\right ) a -a \cos \left (f x +e \right )+\cos \left (f x +e \right ) b -b \right )}{f \sin \left (f x +e \right )^{5} \left (a \cos \left (f x +e \right )+b \right ) b}\)	\(642\)

Verification of antiderivative is not currently implemented for this CAS.

[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*((a*cos(f*x+e)+b)/cos(f*x+e))^(1/2)*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*(2*cos(f*x+e)*EllipticF((-1+cos(
f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+
b))^(1/2)*sin(f*x+e)*b-cos(f*x+e)*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*
x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*a-cos(f*x+e)*EllipticE((-1+cos(f*x+e)
)/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1
)/(a+b))^(1/2)*b+2*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*EllipticF((
-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b*sin(f*x+e)-EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))
^(1/2))*sin(f*x+e)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*a-EllipticE
((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*((a*cos(f*x+e)+b
)/(cos(f*x+e)+1)/(a+b))^(1/2)*b+cos(f*x+e)^2*a-a*cos(f*x+e)+cos(f*x+e)*b-b)/sin(f*x+e)^5/(a*cos(f*x+e)+b)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} A \left (\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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