3.2.99 \(\int \frac {1}{\sqrt {a+b \text {sech}^2(x)}} \, dx\) [199]

Optimal. Leaf size=29 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{\sqrt {a}} \]

[Out]

arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4213, 385, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sech[x]^2],x]

[Out]

ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[a]

Rule 212

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(29)=58\).
time = 0.03, size = 62, normalized size = 2.14 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {a+b+a \sinh ^2(x)}}\right ) \sqrt {a+2 b+a \cosh (2 x)} \text {sech}(x)}{\sqrt {2} \sqrt {a} \sqrt {a+b \text {sech}^2(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sech[x]^2],x]

[Out]

(ArcTanh[(Sqrt[a]*Sinh[x])/Sqrt[a + b + a*Sinh[x]^2]]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[a]*Sq
rt[a + b*Sech[x]^2])

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Maple [F]
time = 1.69, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a +b \mathrm {sech}\left (x \right )^{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*sech(x)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (23) = 46\).
time = 0.40, size = 1059, normalized size = 36.52 \begin {gather*} \left [\frac {\sqrt {a} \log \left (\frac {a b^{2} \cosh \left (x\right )^{8} + 8 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a b^{2} \sinh \left (x\right )^{8} - 2 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )^{6} + 2 \, {\left (14 \, a b^{2} \cosh \left (x\right )^{2} - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{6} + 4 \, {\left (14 \, a b^{2} \cosh \left (x\right )^{3} - 3 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + {\left (a^{3} + 4 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (x\right )^{4} + {\left (70 \, a b^{2} \cosh \left (x\right )^{4} + a^{3} + 4 \, a^{2} b + 9 \, a b^{2} - 30 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (14 \, a b^{2} \cosh \left (x\right )^{5} - 10 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )^{3} + {\left (a^{3} + 4 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + a^{3} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (14 \, a b^{2} \cosh \left (x\right )^{6} - 15 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )^{4} + a^{3} + 3 \, a^{2} b + 3 \, {\left (a^{3} + 4 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (b^{2} \cosh \left (x\right )^{6} + 6 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + b^{2} \sinh \left (x\right )^{6} - 3 \, b^{2} \cosh \left (x\right )^{4} + 3 \, {\left (5 \, b^{2} \cosh \left (x\right )^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (x\right )^{3} - 3 \, b^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - {\left (a^{2} + 4 \, a b\right )} \cosh \left (x\right )^{2} + {\left (15 \, b^{2} \cosh \left (x\right )^{4} - 18 \, b^{2} \cosh \left (x\right )^{2} - a^{2} - 4 \, a b\right )} \sinh \left (x\right )^{2} - a^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{5} - 6 \, b^{2} \cosh \left (x\right )^{3} - {\left (a^{2} + 4 \, a b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a} \sqrt {\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a + 2 \, b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 4 \, {\left (2 \, a b^{2} \cosh \left (x\right )^{7} - 3 \, {\left (a b^{2} - b^{3}\right )} \cosh \left (x\right )^{5} + {\left (a^{3} + 4 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{3} + 3 \, a^{2} b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6}}\right ) + \sqrt {a} \log \left (-\frac {a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, {\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} + a + b\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {a} \sqrt {\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a + 2 \, b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 4 \, {\left (a \cosh \left (x\right )^{3} + {\left (a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + a}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right )}{4 \, a}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {2} {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + a\right )} \sqrt {-a} \sqrt {\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a + 2 \, b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{a b \cosh \left (x\right )^{4} + 4 \, a b \cosh \left (x\right ) \sinh \left (x\right )^{3} + a b \sinh \left (x\right )^{4} - {\left (a^{2} + 3 \, a b\right )} \cosh \left (x\right )^{2} + {\left (6 \, a b \cosh \left (x\right )^{2} - a^{2} - 3 \, a b\right )} \sinh \left (x\right )^{2} - a^{2} + 2 \, {\left (2 \, a b \cosh \left (x\right )^{3} - {\left (a^{2} + 3 \, a b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right ) + \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a + 2 \, b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(a)*log((a*b^2*cosh(x)^8 + 8*a*b^2*cosh(x)*sinh(x)^7 + a*b^2*sinh(x)^8 - 2*(a*b^2 - b^3)*cosh(x)^6 +
2*(14*a*b^2*cosh(x)^2 - a*b^2 + b^3)*sinh(x)^6 + 4*(14*a*b^2*cosh(x)^3 - 3*(a*b^2 - b^3)*cosh(x))*sinh(x)^5 +
(a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^4 + (70*a*b^2*cosh(x)^4 + a^3 + 4*a^2*b + 9*a*b^2 - 30*(a*b^2 - b^3)*cosh(x
)^2)*sinh(x)^4 + 4*(14*a*b^2*cosh(x)^5 - 10*(a*b^2 - b^3)*cosh(x)^3 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x))*sinh(
x)^3 + a^3 + 2*(a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*a*b^2*cosh(x)^6 - 15*(a*b^2 - b^3)*cosh(x)^4 + a^3 + 3*a^2*b
+ 3*(a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*si
nh(x)^6 - 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^
3 - (a^2 + 4*a*b)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 - 4*a*b)*sinh(x)^2 - a^2 + 2*(3*b^2*c
osh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b
)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*a*b^2*cosh(x)^7 - 3*(a*b^2 - b^3)*cosh(x)^5 + (a^3 + 4*a
^2*b + 9*a*b^2)*cosh(x)^3 + (a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*
sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + sqrt(a)*log(
-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(a + b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + b)*sinh(x)^
2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)
/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(a*cosh(x)^3 + (a + b)*cosh(x))*sinh(x) + a)/(cosh(x)^2 + 2*
cosh(x)*sinh(x) + sinh(x)^2)))/a, -1/2*(sqrt(-a)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)
^2 + a)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*b*
cosh(x)^4 + 4*a*b*cosh(x)*sinh(x)^3 + a*b*sinh(x)^4 - (a^2 + 3*a*b)*cosh(x)^2 + (6*a*b*cosh(x)^2 - a^2 - 3*a*b
)*sinh(x)^2 - a^2 + 2*(2*a*b*cosh(x)^3 - (a^2 + 3*a*b)*cosh(x))*sinh(x))) + sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*s
qrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^2 + 2*a*cosh
(x)*sinh(x) + a*sinh(x)^2 + a)))/a]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \operatorname {sech}^{2}{\left (x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*sech(x)**2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/cosh(x)^2)^(1/2),x)

[Out]

int(1/(a + b/cosh(x)^2)^(1/2), x)

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