3.1.12 \(\int \frac {1}{\sqrt {a+b \text {csch}^2(c+d x)}} \, dx\) [12]

Optimal. Leaf size=38 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \text {csch}^2(c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

arctanh(coth(d*x+c)*a^(1/2)/(a+b*csch(d*x+c)^2)^(1/2))/d/a^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4213, 385, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{\sqrt {a} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Csch[c + d*x]^2],x]

[Out]

ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(Sqrt[a]*d)

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(38)=76\).
time = 0.11, size = 97, normalized size = 2.55 \begin {gather*} \frac {\sqrt {-a+2 b+a \cosh (2 (c+d x))} \text {csch}(c+d x) \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )}{\sqrt {2} \sqrt {a} d \sqrt {a+b \text {csch}^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Csch[c + d*x]^2],x]

[Out]

(Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]]*Csch[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cosh[c + d*x] + Sqrt[-a + 2*b + a*Cosh
[2*(c + d*x)]]])/(Sqrt[2]*Sqrt[a]*d*Sqrt[a + b*Csch[c + d*x]^2])

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Maple [F]
time = 2.41, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a +b \mathrm {csch}\left (d x +c \right )^{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a+b*csch(d*x+c)^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*csch(d*x+c)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(b*csch(d*x + c)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (32) = 64\).
time = 0.40, size = 1645, normalized size = 43.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*csch(c + d*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*csch(d*x+c)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex

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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 {sinh}\left (c+d\,x\right )}^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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