\(\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx\) [268]

Optimal result

Integrand size = 23, antiderivative size = 27 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d} \]

[Out]

arcsinh(tan(d*x+c)/(1+sec(d*x+c)))*2^(1/2)/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3892, 221} \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d} \]

[In]

Int[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

Rule 221

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]

Rule 3892

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2} \text {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} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{1+\cos (c+d x)}}}{d} \]

[In]

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

[Out]

(2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]*Sqrt[(1 + Cos[c + d*x])^(-1)])/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs. \(2(25)=50\).

Time = 1.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.44

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {\sec \left (d x +c \right )}\, \sqrt {1+\sec \left (d x +c \right )}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\)	\(93\)

[In]

int(sec(d*x+c)^(1/2)/(1+sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{2 \, d} \]

[In]

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

[Out]

1/2*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))/d

Sympy [F]

\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {\sec {\left (c + d x \right )} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (25) = 50\).

Time = 0.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \log \left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{2 \, d} \]

[In]

integrate(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(c
os(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))/d

Giac [F]

\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {\sec \left (d x + c\right ) + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]

[In]

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

[Out]

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