3.80 \(\int \sqrt {a+a \text {sech}(c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3774, 203} \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sech[c + d*x]],x]

[Out]

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

Rule 203

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

Rule 3774

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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 60, normalized size = 1.62 \[ \frac {\sqrt {2} \sinh ^{-1}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cosh (c+d x)} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\text {sech}(c+d x)+1)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[(c + d*x)/2]]*Sqrt[Cosh[c + d*x]]*Sech[(c + d*x)/2]*Sqrt[a*(1 + Sech[c + d*x])])
/d

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fricas [B]  time = 0.41, size = 637, normalized size = 17.22 \[ \frac {\sqrt {a} \log \left (-\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) - 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} - 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} - 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} - 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) - 7\right )} \sinh \left (d x + c\right )^{2} - 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} - 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} - 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) - 4\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} - 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} - 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) - 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + \sqrt {a} \log \left (\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) + 1\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} + a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(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)*sin
h(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*co
sh(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)*log((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*cos
h(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*cosh(d*x + c) + a)*sinh(d*x + c) + a)/(cosh(d*x + c) + s
inh(d*x + c))))/d

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giac [B]  time = 0.21, size = 83, normalized size = 2.24 \[ \frac {\frac {2 \, a \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \sqrt {a +a \,\mathrm {sech}\left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}\left (d x + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cosh(c + d*x))^(1/2),x)

[Out]

int((a + a/cosh(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}{\left (c + d x \right )} + a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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