3.1.5 \(\int \frac {1}{a+b \coth ^2(c+d x)} \, dx\) [5]

Optimal. Leaf size=46 \[ \frac {x}{a+b}-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b) d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3741, 3756, 211} \begin {gather*} \frac {x}{a+b}-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x]^2)^(-1),x]

[Out]

x/(a + b) - (Sqrt[b]*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*d)

Rule 211

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

Rule 3741

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/(a - b), x] - Dist[b/(a - b), Int[Sec[e
+ f*x]^2/(a + b*Tan[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a, b]

Rule 3756

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 47, normalized size = 1.02 \begin {gather*} \frac {-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a}}+\tanh ^{-1}(\tanh (c+d x))}{(a+b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[c + d*x]^2)^(-1),x]

[Out]

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

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Maple [A]
time = 0.86, size = 71, normalized size = 1.54

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 b +2 a}+\frac {b \arctan \left (\frac {b \coth \left (d x +c \right )}{\sqrt {a b}}\right )}{\left (a +b \right ) \sqrt {a b}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 b +2 a}}{d}\)	\(71\)
default	\(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 b +2 a}+\frac {b \arctan \left (\frac {b \coth \left (d x +c \right )}{\sqrt {a b}}\right )}{\left (a +b \right ) \sqrt {a b}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 b +2 a}}{d}\)	\(71\)
risch	\(\frac {x}{a +b}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-a b}-b}{a +b}\right )}{2 a \left (a +b \right ) d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 a \left (a +b \right ) d}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 56, normalized size = 1.22 \begin {gather*} \frac {b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a + b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )} d} + \frac {d x + c}{{\left (a + b\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (38) = 76\).
time = 0.41, size = 488, normalized size = 10.61 \begin {gather*} \left [\frac {2 \, d x + \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 6 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + a b\right )} \sqrt {-\frac {b}{a}}}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right )}{2 \, {\left (a + b\right )} d}, \frac {d x - \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a + b\right )} \sinh \left (d x + c\right )^{2} - a + b\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right )}{{\left (a + b\right )} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*d*x + sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d
*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 - 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cos
h(d*x + c)^2 - a^2 + b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2
- b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x +
c) + (a^2 + a*b)*sinh(d*x + c)^2 - a^2 + a*b)*sqrt(-b/a))/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*s
inh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - a + b)*s
inh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)))/((a + b)*d), (d*
x - sqrt(b/a)*arctan(1/2*((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 + b)*sqrt(b/a)/b))/((a + b)*d)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (37) = 74\).
time = 3.90, size = 253, normalized size = 5.50 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x}{\coth ^{2}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x - \frac {\tanh {\left (c + d x \right )}}{d}}{b} & \text {for}\: a = 0 \\- \frac {d x \tanh ^{2}{\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} + \frac {d x}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} - \frac {\tanh {\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} & \text {for}\: a = - b \\\frac {x}{a} & \text {for}\: b = 0 \\\frac {x}{a + b \coth ^{2}{\left (c \right )}} & \text {for}\: d = 0 \\\frac {2 a d x \sqrt {- \frac {b}{a}}}{2 a^{2} d \sqrt {- \frac {b}{a}} + 2 a b d \sqrt {- \frac {b}{a}}} - \frac {b \log {\left (- \sqrt {- \frac {b}{a}} + \tanh {\left (c + d x \right )} \right )}}{2 a^{2} d \sqrt {- \frac {b}{a}} + 2 a b d \sqrt {- \frac {b}{a}}} + \frac {b \log {\left (\sqrt {- \frac {b}{a}} + \tanh {\left (c + d x \right )} \right )}}{2 a^{2} d \sqrt {- \frac {b}{a}} + 2 a b d \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x/coth(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((x - tanh(c + d*x)/d)/b, Eq(a, 0)), (-d*x*tanh(
c + d*x)**2/(2*b*d*tanh(c + d*x)**2 - 2*b*d) + d*x/(2*b*d*tanh(c + d*x)**2 - 2*b*d) - tanh(c + d*x)/(2*b*d*tan
h(c + d*x)**2 - 2*b*d), Eq(a, -b)), (x/a, Eq(b, 0)), (x/(a + b*coth(c)**2), Eq(d, 0)), (2*a*d*x*sqrt(-b/a)/(2*
a**2*d*sqrt(-b/a) + 2*a*b*d*sqrt(-b/a)) - b*log(-sqrt(-b/a) + tanh(c + d*x))/(2*a**2*d*sqrt(-b/a) + 2*a*b*d*sq
rt(-b/a)) + b*log(sqrt(-b/a) + tanh(c + d*x))/(2*a**2*d*sqrt(-b/a) + 2*a*b*d*sqrt(-b/a)), True))

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Giac [A]
time = 0.40, size = 65, normalized size = 1.41 \begin {gather*} -\frac {\frac {b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )}} - \frac {d x + c}{a + b}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.11, size = 37, normalized size = 0.80 \begin {gather*} \frac {x}{a+b}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {coth}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )}{d\,\sqrt {a\,b}\,\left (a+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*coth(c + d*x)^2),x)

[Out]

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

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