3.63 \(\int \frac{\sqrt{a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx\)

Optimal. Leaf size=92 \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]

[Out]

(EllipticE[ArcSin[Tan[c + d*x]/(1 + Sec[c + d*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[a + b*
Sec[c + d*x]])/(d*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])

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Rubi [A]  time = 0.164096, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2829, 3968} \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sec[c + d*x]]/(1 + Cos[c + d*x]),x]

[Out]

(EllipticE[ArcSin[Tan[c + d*x]/(1 + Sec[c + d*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[a + b*
Sec[c + d*x]])/(d*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])

Rule 2829

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int
[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !In
tegerQ[n] && IntegerQ[m]

Rule 3968

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx &=\int \frac{\sec (c+d x) \sqrt{a+b \sec (c+d x)}}{1+\sec (c+d x)} \, dx\\ &=\frac{E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )|\frac{a-b}{a+b}\right ) \sqrt{\frac{1}{1+\sec (c+d x)}} \sqrt{a+b \sec (c+d x)}}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (1+\sec (c+d x))}}}\\ \end{align*}

Mathematica [A]  time = 1.56502, size = 85, normalized size = 0.92 \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sec[c + d*x]]/(1 + Cos[c + d*x]),x]

[Out]

(EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[a + b*Sec[c + d*x]])/
(d*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])

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Maple [A]  time = 0.305, size = 150, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -a-b \right ) \left ( \cos \left ( dx+c \right ) -1 \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}{d \left ( a\cos \left ( dx+c \right ) +b \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{\it EllipticE} \left ({\frac{\cos \left ( dx+c \right ) -1}{\sin \left ( dx+c \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) \sqrt{{\frac{a\cos \left ( dx+c \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{a\cos \left ( dx+c \right ) +b}{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*(-a-b)*EllipticE((cos(d*x+c)-1)/sin(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(a*cos(d*x+c)+b)/(1+cos(d*x+c)))
^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)-1)*((a*cos(d*x+c)+b)/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^2/(
a*cos(d*x+c)+b)/sin(d*x+c)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c) + a)/(cos(d*x + c) + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)/(cos(d*x + c) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sec(c + d*x))/(cos(c + d*x) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c) + a)/(cos(d*x + c) + 1), x)