3.217 \(\int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx\)

Optimal. Leaf size=652 \[ \frac{2 (b c-a d) \cot (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(b c-a d) (\sec (e+f x)-1)}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right ),\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a b f \sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}}-\frac{2 c (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{\frac{(a+b) (b c-a d) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt{-\frac{(a+b) (a d-b c) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\sqrt{\frac{(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f (a+b) \sqrt{c+d \sec (e+f x)}} \]

[Out]

(-2*c*(c + d)*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c
+ d)*(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]))]*(a + b*Sec[e + f*x])^(3/2)*Sqrt[((a + b)*(b*c - a*d)*(-1 + Sec[e + f*x])*(c + d*Sec
[e + f*x]))/((c + d)^2*(a + b*Sec[e + f*x])^2)])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]) + (2*d*(c + d)*Cot[e +
f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(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]))]*(a + b*Sec[e + f*x])^(3/2)*Sqrt[-(((a + b)*(-(b*c) + a*d)*(-1 + Sec[e + f*x])*(c + d*Sec[e + f*x]))/((c
+ d)^2*(a + b*Sec[e + f*x])^2))])/(b*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*Cot[e + f*x]*Ellipti
cF[ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(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]))]*Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a*b*f*Sqrt[((a + b)*
(c + d*Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))])

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Rubi [F]  time = 0.092558, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

Defer[Int][(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]], x]

Rubi steps

\begin{align*} \int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx &=\int \frac{(c+d \sec (e+f x))^{3/2}}{\sqrt{a+b \sec (e+f x)}} \, dx\\ \end{align*}

Mathematica [C]  time = 32.6576, size = 49385, normalized size = 75.74 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

Result too large to show

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Maple [A]  time = 0.386, size = 491, normalized size = 0.8 \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 ( 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 ){c}^{2}+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 ){d}^{2}-{\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 ){c}^{2}+2\,{\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 ) cd-{\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 ){d}^{2} \right ) \sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\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((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x)

[Out]

2/f/((a-b)/(a+b))^(1/2)*(2*EllipticPi((-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))*c^2+2*EllipticPi((-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))*d^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))*c^2+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))*c*d-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))*d^2)*cos(f
*x+e)*sin(f*x+e)^2*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*(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{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{\sqrt{b \sec \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*sec(f*x + e) + c)^(3/2)/sqrt(b*sec(f*x + e) + a), 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((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((c+d*sec(f*x+e))**(3/2)/(a+b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{\sqrt{b \sec \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e) + c)^(3/2)/sqrt(b*sec(f*x + e) + a), x)