∫ 1__________∘ a + b coth(c + dx) dx [88]

3.1.88 \(\int \frac {1}{\sqrt {a+b \coth (c+d x)}} \, dx\) [88]

Optimal. Leaf size=74 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d} \]

[Out]

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

b)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3566, 722, 1107, 213} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[a + b*Coth[c + d*x]],x]

[Out]

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

b]]/(Sqrt[a + b]*d)

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]

, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c

*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/

2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*

a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x

, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b \coth (c+d x)}} \, dx &=-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {(2 b) \text {Subst}\left (\int \frac {1}{a^2-b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \coth (c+d x)}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{-a-b+x^2} \, dx,x,\sqrt {a+b \coth (c+d x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \coth (c+d x)}\right )}{d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.00, size = 129, normalized size = 1.74 \begin {gather*} -\frac {\left (\frac {\tanh ^{-1}\left (\frac {\sqrt {i (a+b \coth (c+d x))}}{\sqrt {i (a-b)}}\right )}{\sqrt {i (a-b)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {i (a+b \coth (c+d x))}}{\sqrt {i (a+b)}}\right )}{\sqrt {i (a+b)}}\right ) \sqrt {i (a+b \coth (c+d x))}}{d \sqrt {a+b \coth (c+d x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

x])]/Sqrt[I*(a + b)]]/Sqrt[I*(a + b)])*Sqrt[I*(a + b*Coth[c + d*x])])/(d*Sqrt[a + b*Coth[c + d*x]]))

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Maple [A]
time = 1.79, size = 70, normalized size = 0.95


method	result	size

derivativedivides	\(-\frac {2 b \left (-\frac {\arctanh \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {a +b}}\right )}{2 b \sqrt {a +b}}-\frac {\arctan \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{2 b \sqrt {-a +b}}\right )}{d}\)	\(70\)
default	\(-\frac {2 b \left (-\frac {\arctanh \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {a +b}}\right )}{2 b \sqrt {a +b}}-\frac {\arctan \left (\frac {\sqrt {a +b \coth \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{2 b \sqrt {-a +b}}\right )}{d}\)	\(70\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

x+c))^(1/2)/(-a+b)^(1/2)))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (62) = 124\).
time = 0.45, size = 2307, normalized size = 31.18 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(a + b)*(a - b)*log(2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh

(d*x + c)^3 + 2*(a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 - 4*(a^2 + a*b)*cosh(d*x + c)^2 + 4*(3*(a^2 + 2*a*b + b^2)

*cosh(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c)^2 + 2*a^2 - b^2 + 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 - (2*a + b)*cosh(d*x + c)^2 + (6*(a + b)*cosh(d*x + c)^2 - 2*a

- b)*sinh(d*x + c)^2 + 2*(2*(a + b)*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a + b)*

sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c)) + 8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 + a*b)

*cosh(d*x + c))*sinh(d*x + c)) + (a + b)*sqrt(a - b)*log(((2*a^2 - b^2)*cosh(d*x + c)^4 + 4*(2*a^2 - b^2)*cosh

(d*x + c)*sinh(d*x + c)^3 + (2*a^2 - b^2)*sinh(d*x + c)^4 - 4*(a^2 - a*b)*cosh(d*x + c)^2 + 2*(3*(2*a^2 - b^2)

*cosh(d*x + c)^2 - 2*a^2 + 2*a*b)*sinh(d*x + c)^2 + 2*a^2 - 4*a*b + 2*b^2 - 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 - b)*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 - 2*a + b)*sinh(

d*x + c)^2 + 2*(2*a*cosh(d*x + c)^3 - (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)*sqrt(a - b)*sqrt((b*cosh

(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c)) + 4*((2*a^2 - b^2)*cosh(d*x + c)^3 - 2*(a^2 - a*b)*cosh(d*x + c))*

sinh(d*x + c))/(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh

(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)))/((a^2 - b^2)*d), -1/4*(2*(a - b)*sqrt(-a - b)*arctan(((a + b)*c

osh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 - a)*sqrt(-a - b)*sqrt((b*cos

h(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c))/((a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 2*(a^2 + 2*a*b + b^2)*cosh

(d*x + c)*sinh(d*x + c) + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 - a^2 + b^2)) - (a + b)*sqrt(a - b)*log(((2*a^2

- b^2)*cosh(d*x + c)^4 + 4*(2*a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 - b^2)*sinh(d*x + c)^4 - 4*(a^

2 - a*b)*cosh(d*x + c)^2 + 2*(3*(2*a^2 - b^2)*cosh(d*x + c)^2 - 2*a^2 + 2*a*b)*sinh(d*x + c)^2 + 2*a^2 - 4*a*b

+ 2*b^2 - 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 - b)*cosh(d*x +

c)^2 + (6*a*cosh(d*x + c)^2 - 2*a + b)*sinh(d*x + c)^2 + 2*(2*a*cosh(d*x + c)^3 - (2*a - b)*cosh(d*x + c))*si

nh(d*x + c) + a - b)*sqrt(a - b)*sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c)) + 4*((2*a^2 - b^2)*co

sh(d*x + c)^3 - 2*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c)

+ 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)))/((a^2 - b^2)*d), -

1/4*(2*(a + b)*sqrt(-a + b)*arctan(-(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 -

a + b)*sqrt(-a + b)*sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c))/((a^2 - b^2)*cosh(d*x + c)^2 + 2*

(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*sinh(d*x + c)^2 - a^2 + 2*a*b - b^2)) - sqrt(a + b)*(a -

b)*log(2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a^2 +

2*a*b + b^2)*sinh(d*x + c)^4 - 4*(a^2 + a*b)*cosh(d*x + c)^2 + 4*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - a^2

- a*b)*sinh(d*x + c)^2 + 2*a^2 - b^2 + 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 - (2*a + b)*cosh(d*x + c)^2 + (6*(a + b)*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x + c)^2 +

2*(2*(a + b)*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a + b)*sqrt((b*cosh(d*x + c)

+ a*sinh(d*x + c))/sinh(d*x + c)) + 8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 + a*b)*cosh(d*x + c))*sinh(d

*x + c)))/((a^2 - b^2)*d), -1/2*((a + b)*sqrt(-a + b)*arctan(-(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x

+ c) + a*sinh(d*x + c)^2 - a + b)*sqrt(-a + b)*sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c))/((a^2 -

b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*sinh(d*x + c)^2 - a^2 + 2*a*b

- b^2)) + (a - b)*sqrt(-a - b)*arctan(((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a +

b)*sinh(d*x + c)^2 - a)*sqrt(-a - b)*sqrt((b*cosh(d*x + c) + a*sinh(d*x + c))/sinh(d*x + c))/((a^2 + 2*a*b + b

^2)*cosh(d*x + c)^2 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^2

- a^2 + b^2)))/((a^2 - b^2)*d)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*coth(c + d*x)), 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*coth(d*x+c))^(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 [B]
time = 1.47, size = 242, normalized size = 3.27 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}}{\left (\frac {16\,b^4\,d^3}{a\,d^3-b\,d^3}-\frac {16\,a\,b^3\,d^3}{a\,d^3-b\,d^3}\right )\,\sqrt {a-b}}+\frac {\left (a\,d^3-b\,d^3\right )\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}}{b\,d^3\,\sqrt {a-b}}\right )}{d\,\sqrt {a-b}}-\frac {\mathrm {atanh}\left (\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}}{\left (\frac {16\,b^4\,d^3}{a\,d^3+b\,d^3}+\frac {16\,a\,b^3\,d^3}{a\,d^3+b\,d^3}\right )\,\sqrt {a+b}}-\frac {\left (a\,d^3+b\,d^3\right )\,\sqrt {a+b\,\mathrm {coth}\left (c+d\,x\right )}}{b\,d^3\,\sqrt {a+b}}\right )}{d\,\sqrt {a+b}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh((16*a*b^2*(a + b*coth(c + d*x))^(1/2))/(((16*b^4*d^3)/(a*d^3 - b*d^3) - (16*a*b^3*d^3)/(a*d^3 - b*d^3))*

(a - b)^(1/2)) + ((a*d^3 - b*d^3)*(a + b*coth(c + d*x))^(1/2))/(b*d^3*(a - b)^(1/2)))/(d*(a - b)^(1/2)) - atan

h((16*a*b^2*(a + b*coth(c + d*x))^(1/2))/(((16*b^4*d^3)/(a*d^3 + b*d^3) + (16*a*b^3*d^3)/(a*d^3 + b*d^3))*(a +

b)^(1/2)) - ((a*d^3 + b*d^3)*(a + b*coth(c + d*x))^(1/2))/(b*d^3*(a + b)^(1/2)))/(d*(a + b)^(1/2))

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