3.269 \(\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)}} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d} \]

[Out]

-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Sec[
c + d*x]])

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Rubi [A]  time = 0.078271, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3812, 3807, 215} \[ \frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Sec[
c + d*x]])

Rule 3812

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(Cot[
e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + 1)), x] + Dist[(a*m)/(b*d*(m + 1)), Int[(a + b*Csc
[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m
+ n + 1, 0] &&  !LtQ[m, -2^(-1)]

Rule 3807

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)}} \, dx &=\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}-\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,-\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}\\ &=-\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.207902, size = 90, normalized size = 1.45 \[ \frac{2 \sin (c+d x) \sqrt{-(\sec (c+d x)-1) \sec (c+d x)}+\sqrt{2} \tan (c+d x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right )}{d \sqrt{-\tan ^2(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1
- Sec[c + d*x]]]*Tan[c + d*x])/(d*Sqrt[-Tan[c + d*x]^2])

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Maple [A]  time = 0.155, size = 96, normalized size = 1.6 \begin{align*}{\frac{1}{d\sin \left ( dx+c \right ) } \left ( \arctan \left ({\frac{\sin \left ( dx+c \right ) }{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) -2\,\cos \left ( dx+c \right ) +2 \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) +1}{\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/sec(d*x+c)^(1/2)/(1+sec(d*x+c))^(1/2),x)

[Out]

1/d*(arctan(1/2*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2))*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-2*cos(d*x+c)+2)*((c
os(d*x+c)+1)/cos(d*x+c))^(1/2)/(1/cos(d*x+c))^(1/2)/sin(d*x+c)

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Maxima [A]  time = 2.20209, size = 136, normalized size = 2.19 \begin{align*} -\frac{\sqrt{2} \log \left (\cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) - \sqrt{2} \log \left (\cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) - 4 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*log(cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c) + 1) - sqrt(2)*log(
cos(1/2*d*x + 1/2*c)^2 + sin(1/2*d*x + 1/2*c)^2 - 2*sin(1/2*d*x + 1/2*c) + 1) - 4*sqrt(2)*sin(1/2*d*x + 1/2*c)
)/d

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Fricas [B]  time = 1.93437, size = 404, normalized size = 6.52 \begin{align*} \frac{{\left (\sqrt{2} \cos \left (d x + c\right ) + \sqrt{2}\right )} \log \left (-\frac{2 \, \sqrt{2} \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((sqrt(2)*cos(d*x + c) + sqrt(2))*log(-(2*sqrt(2)*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))
*sin(d*x + c) + cos(d*x + c)^2 - 2*cos(d*x + c) - 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*sqrt((cos(d*x
+ c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(sec(c + d*x) + 1)*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{\sec \left (d x + c\right ) + 1} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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