### 3.366 $$\int \tanh ^2(a+b x) \, dx$$

Optimal. Leaf size=13 $x-\frac{\tanh (a+b x)}{b}$

[Out]

x - Tanh[a + b*x]/b

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Rubi [A]  time = 0.0093024, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3473, 8} $x-\frac{\tanh (a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[a + b*x]^2,x]

[Out]

x - Tanh[a + b*x]/b

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0068711, size = 23, normalized size = 1.77 $\frac{\tanh ^{-1}(\tanh (a+b x))}{b}-\frac{\tanh (a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[a + b*x]^2,x]

[Out]

ArcTanh[Tanh[a + b*x]]/b - Tanh[a + b*x]/b

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Maple [A]  time = 0.013, size = 18, normalized size = 1.4 \begin{align*}{\frac{bx+a-\tanh \left ( bx+a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(b*x+a-tanh(b*x+a))

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Maxima [A]  time = 1.04342, size = 34, normalized size = 2.62 \begin{align*} x + \frac{a}{b} - \frac{2}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.97179, size = 82, normalized size = 6.31 \begin{align*} \frac{{\left (b x + 1\right )} \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + 1)*cosh(b*x + a) - sinh(b*x + a))/(b*cosh(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh ^{2}{\left (a + b x \right )} \operatorname{sech}^{2}{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**2*sinh(b*x+a)**2,x)

[Out]

Integral(sinh(a + b*x)**2*sech(a + b*x)**2, x)

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Giac [B]  time = 1.20453, size = 38, normalized size = 2.92 \begin{align*} \frac{b x + a}{b} + \frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)/b + 2/(b*(e^(2*b*x + 2*a) + 1))