\(\int \frac {1}{a+b \tanh ^2(c+d x)} \, dx\) [175]

Optimal result

Integrand size = 14, antiderivative size = 45 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\frac {x}{a+b}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3741, 3756, 211} \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)}+\frac {x}{a+b} \]

[In]

Int[(a + b*Tanh[c + d*x]^2)^(-1),x]

[Out]

x/(a + b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(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*} \text {integral}& = \frac {x}{a+b}+\frac {b \int \frac {\text {sech}^2(c+d x)}{a+b \tanh ^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,\tanh (c+d x)\right )}{(a+b) d} \\ & = \frac {x}{a+b}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}-\log (1-\tanh (c+d x))+\log (1+\tanh (c+d x))}{2 a d+2 b d} \]

[In]

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

[Out]

((2*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] - Log[1 - Tanh[c + d*x]] + Log[1 + Tanh[c + d*x]]
)/(2*a*d + 2*b*d)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.58

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (37) = 74\).

Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 10.76 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\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 ] \]

[In]

integrate(1/(a+b*tanh(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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (37) = 74\).

Time = 2.00 (sec) , antiderivative size = 240, normalized size of antiderivative = 5.33 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\begin {cases} \frac {\tilde {\infty } x}{\tanh ^{2}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x}{a} & \text {for}\: b = 0 \\\frac {x - \frac {1}{d \tanh {\left (c + d x \right )}}}{b} & \text {for}\: a = 0 \\- \frac {d x \tanh ^{2}{\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} + \frac {d x}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} + \frac {\tanh {\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} & \text {for}\: a = - b \\\frac {x}{a + b \tanh ^{2}{\left (c \right )}} & \text {for}\: d = 0 \\\frac {2 d x \sqrt {- \frac {a}{b}}}{2 a d \sqrt {- \frac {a}{b}} + 2 b d \sqrt {- \frac {a}{b}}} + \frac {\log {\left (- \sqrt {- \frac {a}{b}} + \tanh {\left (c + d x \right )} \right )}}{2 a d \sqrt {- \frac {a}{b}} + 2 b d \sqrt {- \frac {a}{b}}} - \frac {\log {\left (\sqrt {- \frac {a}{b}} + \tanh {\left (c + d x \right )} \right )}}{2 a d \sqrt {- \frac {a}{b}} + 2 b d \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*x/tanh(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x/a, Eq(b, 0)), ((x - 1/(d*tanh(c + d*x)))/b, E
q(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) + tan
h(c + d*x)/(2*b*d*tanh(c + d*x)**2 - 2*b*d), Eq(a, -b)), (x/(a + b*tanh(c)**2), Eq(d, 0)), (2*d*x*sqrt(-a/b)/(
2*a*d*sqrt(-a/b) + 2*b*d*sqrt(-a/b)) + log(-sqrt(-a/b) + tanh(c + d*x))/(2*a*d*sqrt(-a/b) + 2*b*d*sqrt(-a/b))
- log(sqrt(-a/b) + tanh(c + d*x))/(2*a*d*sqrt(-a/b) + 2*b*d*sqrt(-a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=-\frac {b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )} d} + \frac {d x + c}{{\left (a + b\right )} d} \]

[In]

integrate(1/(a+b*tanh(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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.40 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )}} + \frac {d x + c}{a + b}}{d} \]

[In]

integrate(1/(a+b*tanh(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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a+b \tanh ^2(c+d x)} \, dx=\frac {x}{a+b}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tanh}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )}{d\,\sqrt {a\,b}\,\left (a+b\right )} \]

[In]

int(1/(a + b*tanh(c + d*x)^2),x)

[Out]

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