3.676 \(\int \frac{1}{\sqrt{-3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{4 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x+\pi ),6\right )}{3 d \sqrt{-2 \sec (c+d x)-3}}-\frac{2 \sqrt{-2 \sec (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}} \]

[Out]

(-2*EllipticE[(c + Pi + d*x)/2, 6]*Sqrt[-3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]
]) - (4*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 - 2*Sec[c +
d*x]])

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Rubi [A]  time = 0.174563, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3862, 3856, 2654, 3858, 2662} \[ -\frac{4 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt{-2 \sec (c+d x)-3}}-\frac{2 \sqrt{-2 \sec (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]

[Out]

(-2*EllipticE[(c + Pi + d*x)/2, 6]*Sqrt[-3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]
]) - (4*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 - 2*Sec[c +
d*x]])

Rule 3862

Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[1/a,
Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[b/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b
*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3856

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2654

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a - b]*EllipticE[(1*(c + Pi/2 + d*x)
)/2, (-2*b)/(a - b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2662

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c + Pi/2 + d*x))/2, (-2*b
)/(a - b)])/(d*Sqrt[a - b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx &=-\left (\frac{1}{3} \int \frac{\sqrt{-3-2 \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\right )-\frac{2}{3} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3-2 \sec (c+d x)}} \, dx\\ &=-\frac{\sqrt{-3-2 \sec (c+d x)} \int \sqrt{-2-3 \cos (c+d x)} \, dx}{3 \sqrt{-2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (2 \sqrt{-2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{-2-3 \cos (c+d x)}} \, dx}{3 \sqrt{-3-2 \sec (c+d x)}}\\ &=-\frac{2 E\left (\left .\frac{1}{2} (c+\pi +d x)\right |6\right ) \sqrt{-3-2 \sec (c+d x)}}{3 d \sqrt{-2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{4 \sqrt{-2-3 \cos (c+d x)} F\left (\left .\frac{1}{2} (c+\pi +d x)\right |6\right ) \sqrt{\sec (c+d x)}}{3 d \sqrt{-3-2 \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.104836, size = 81, normalized size = 0.7 \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \left (5 E\left (\frac{1}{2} (c+d x)|\frac{6}{5}\right )-2 \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )\right )}{3 \sqrt{5} d \sqrt{-2 \sec (c+d x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]

[Out]

(2*Sqrt[2 + 3*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 6/5] - 2*EllipticF[(c + d*x)/2, 6/5])*Sqrt[Sec[c + d*x]]
)/(3*Sqrt[5]*d*Sqrt[-3 - 2*Sec[c + d*x]])

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Maple [C]  time = 0.255, size = 394, normalized size = 3.4 \begin{align*} -{\frac{1}{15\,d\sin \left ( dx+c \right ) \left ( 2+3\,\cos \left ( dx+c \right ) \right ) } \left ( 3\,i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-5\,i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}+3\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -5\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{10}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -30\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+10\,\cos \left ( dx+c \right ) +20 \right ) \sqrt{-{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x)

[Out]

-1/15/d*(3*I*sin(d*x+c)*cos(d*x+c)*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)*(1/(cos(d*x+c
)+1))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)-5*I*sin(d*x+c)*cos(d*x+c)*EllipticE(I*(-1+cos(d*x
+c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)+3*I*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*10^(1/2)*((2+3*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-5*I*EllipticE(I*(-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*2^(1/2)*
(1/(cos(d*x+c)+1))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-30*cos(d*x+c)^2+10*cos(d*
x+c)+20)*(-(2+3*cos(d*x+c))/cos(d*x+c))^(1/2)/(1/cos(d*x+c))^(1/2)/sin(d*x+c)/(2+3*cos(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-2*sec(d*x + c) - 3)*sqrt(sec(d*x + c))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right )^{2} + 3 \, \sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-2*sec(d*x + c) - 3)*sqrt(sec(d*x + c))/(2*sec(d*x + c)^2 + 3*sec(d*x + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 \sec{\left (c + d x \right )} - 3} \sqrt{\sec{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3-2*sec(d*x+c))**(1/2)/sec(d*x+c)**(1/2),x)

[Out]

Integral(1/(sqrt(-2*sec(c + d*x) - 3)*sqrt(sec(c + d*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-2*sec(d*x + c) - 3)*sqrt(sec(d*x + c))), x)