3.265 \(\int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx\)

Optimal. Leaf size=196 \[ \frac{2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt{\frac{a+b}{c+d}}} \]

[Out]

(2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sq
rt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)
*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e +
f*x]))/(d*Sqrt[(a + b)/(c + d)]*f)

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Rubi [A]  time = 0.192992, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3982} \[ \frac{2 \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt{\frac{a+b}{c+d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sq
rt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)
*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e +
f*x]))/(d*Sqrt[(a + b)/(c + d)]*f)

Rule 3982

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Simp[(-2*(a + b*Csc[e + f*x])*Sqrt[-(((b*c - a*d)*(1 - Csc[e + f*x]))/((c + d)*(a + b*Csc[
e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Csc[e + f*x]))/((c - d)*(a + b*Csc[e + f*x]))]*EllipticPi[(b*(c + d))/(d*(
a + b)), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]]], ((a - b)*(c + d))/
((a + b)*(c - d))])/(d*f*Sqrt[(a + b)/(c + d)]*Cot[e + f*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]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) \sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx &=\frac{2 \cot (e+f x) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{\frac{a+b}{c+d}} \sqrt{c+d \sec (e+f x)}}{\sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{d \sqrt{\frac{a+b}{c+d}} f}\\ \end{align*}

Mathematica [C]  time = 32.4728, size = 44216, normalized size = 225.59 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [A]  time = 0.404, size = 351, normalized size = 1.8 \begin{align*} 2\,{\frac{\cos \left ( fx+e \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{f \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},\sqrt{{\frac{ \left ( c-d \right ) \left ( a+b \right ) }{ \left ( a-b \right ) \left ( c+d \right ) }}} \right ) a-{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},\sqrt{{\frac{ \left ( c-d \right ) \left ( a+b \right ) }{ \left ( a-b \right ) \left ( c+d \right ) }}} \right ) b+2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},{\frac{a+b}{a-b}},{\sqrt{{\frac{c-d}{c+d}}}{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \right ) b \right ) \sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2),x)

[Out]

2/f/((a-b)/(a+b))^(1/2)*(EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1
/2))*a-EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b+2*EllipticP
i((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*b)*cos(f
*x+e)*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*sin(f*x+e)^2*(1/(c+d)*(d+c*cos
(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)/(-1+cos(f*x+e))/(d+c*cos(f*x+e)
)/(a*cos(f*x+e)+b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{\sqrt{d \sec \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(f*x + e) + a)*sec(f*x + e)/sqrt(d*sec(f*x + e) + c), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sec{\left (e + f x \right )}} \sec{\left (e + f x \right )}}{\sqrt{c + d \sec{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+b*sec(f*x+e))**(1/2)/(c+d*sec(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a + b*sec(e + f*x))*sec(e + f*x)/sqrt(c + d*sec(e + f*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{\sqrt{d \sec \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(f*x + e) + a)*sec(f*x + e)/sqrt(d*sec(f*x + e) + c), x)