\(\int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx\) [81]

Optimal result

Integrand size = 14, antiderivative size = 85 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

2*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/d/a^(1/2)-arctanh(1/2*a^(1/2)*tanh(d*x+c)*2^(1/2)/(a+a*
sech(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3861, 3859, 209, 3880} \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d} \]

[In]

Int[1/Sqrt[a + a*Sech[c + d*x]],x]

[Out]

(2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Tanh[c
+ d*x])/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(Sqrt[a]*d)

Rule 209

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

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3861

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

Rule 3880

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {a+a \text {sech}(c+d x)} \, dx}{a}-\int \frac {\text {sech}(c+d x)}{\sqrt {a+a \text {sech}(c+d x)}} \, dx \\ & = \frac {(2 i) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {\left (1+e^{c+d x}\right ) \left (\sqrt {2} \text {arcsinh}\left (e^{c+d x}\right )-2 \text {arctanh}\left (\frac {-1+e^{c+d x}}{\sqrt {2} \sqrt {1+e^{2 (c+d x)}}}\right )-\sqrt {2} \text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )\right )}{\sqrt {2} d \sqrt {1+e^{2 (c+d x)}} \sqrt {a (1+\text {sech}(c+d x))}} \]

[In]

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

[Out]

((1 + E^(c + d*x))*(Sqrt[2]*ArcSinh[E^(c + d*x)] - 2*ArcTanh[(-1 + E^(c + d*x))/(Sqrt[2]*Sqrt[1 + E^(2*(c + d*
x))])] - Sqrt[2]*ArcTanh[Sqrt[1 + E^(2*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^(2*(c + d*x))]*Sqrt[a*(1 + Sech[c
+ d*x])])

Maple [F]

\[\int \frac {1}{\sqrt {a +\operatorname {sech}\left (d x +c \right ) a}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (70) = 140\).

Time = 0.28 (sec) , antiderivative size = 868, normalized size of antiderivative = 10.21 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(sqrt(2)*sqrt(a)*log(-(3*cosh(d*x + c)^2 + 2*(3*cosh(d*x + c) - 1)*sinh(d*x + c) + 3*sinh(d*x + c)^2 - 2*s
qrt(2)*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(
d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)
*sinh(d*x + c) + sinh(d*x + c)^2 + 1))/sqrt(a) - 2*cosh(d*x + c) + 3)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)
*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)) + sqrt(a)*log(-(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4
- 3*a*cosh(d*x + c)^3 + (4*a*cosh(d*x + c) - 3*a)*sinh(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^
2 - 9*a*cosh(d*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + c) - 3)*sinh(d*x + c)^4 + sinh
(d*x + c)^5 - 3*cosh(d*x + c)^4 + (10*cosh(d*x + c)^2 - 12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c
)^3 + (10*cosh(d*x + c)^3 - 18*cosh(d*x + c)^2 + 15*cosh(d*x + c) - 7)*sinh(d*x + c)^2 - 7*cosh(d*x + c)^2 + (
5*cosh(d*x + c)^4 - 12*cosh(d*x + c)^3 + 15*cosh(d*x + c)^2 - 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x
+ c) - 4)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - 4*a*cosh(
d*x + c) + (4*a*cosh(d*x + c)^3 - 9*a*cosh(d*x + c)^2 + 10*a*cosh(d*x + c) - 4*a)*sinh(d*x + c) + 4*a)/(cosh(d
*x + c)^3 + 3*cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3)) + sqrt(a)*lo
g((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + (cosh(d*x + c)^3 + (3*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + sinh(d*x
+ c)^3 + cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) + 1)*sqrt(
a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + a*cosh(d*x + c) + (2*a*co
sh(d*x + c) + a)*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c))))/(a*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (70) = 140\).

Time = 0.65 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {\frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} e^{\left (-d x\right )} + \sqrt {a} e^{c} - \sqrt {a e^{\left (-2 \, d x\right )} + a e^{\left (2 \, c\right )}}\right )} e^{\left (-c\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} + \frac {\log \left ({\left | -\sqrt {a} e^{\left (-d x\right )} + \sqrt {a} e^{c} + \sqrt {a e^{\left (-2 \, d x\right )} + a e^{\left (2 \, c\right )}} \right |}\right )}{\sqrt {a}} - \frac {\log \left ({\left | -\sqrt {a} e^{\left (-d x\right )} - \sqrt {a} e^{c} + \sqrt {a e^{\left (-2 \, d x\right )} + a e^{\left (2 \, c\right )}} \right |}\right )}{\sqrt {a}} + \frac {\log \left ({\left | -\sqrt {a} e^{\left (-d x\right )} + \sqrt {a e^{\left (-2 \, d x\right )} + a e^{\left (2 \, c\right )}} \right |}\right )}{\sqrt {a}}}{d} \]

[In]

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

[Out]

(2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(a)*e^(-d*x) + sqrt(a)*e^c - sqrt(a*e^(-2*d*x) + a*e^(2*c)))*e^(-c)/sqrt(-
a))/sqrt(-a) + log(abs(-sqrt(a)*e^(-d*x) + sqrt(a)*e^c + sqrt(a*e^(-2*d*x) + a*e^(2*c))))/sqrt(a) - log(abs(-s
qrt(a)*e^(-d*x) - sqrt(a)*e^c + sqrt(a*e^(-2*d*x) + a*e^(2*c))))/sqrt(a) + log(abs(-sqrt(a)*e^(-d*x) + sqrt(a*
e^(-2*d*x) + a*e^(2*c))))/sqrt(a))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]

[In]

int(1/(a + a/cosh(c + d*x))^(1/2),x)

[Out]

int(1/(a + a/cosh(c + d*x))^(1/2), x)