### 3.84 $$\int \text{sech}^2(a+b x) \tanh ^2(a+b x) \, dx$$

Optimal. Leaf size=15 $\frac{\tanh ^3(a+b x)}{3 b}$

[Out]

Tanh[a + b*x]^3/(3*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0289947, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2607, 30} $\frac{\tanh ^3(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[a + b*x]^2*Tanh[a + b*x]^2,x]

[Out]

Tanh[a + b*x]^3/(3*b)

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \text{sech}^2(a+b x) \tanh ^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int x^2 \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{\tanh ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0062168, size = 15, normalized size = 1. $\frac{\tanh ^3(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[a + b*x]^2*Tanh[a + b*x]^2,x]

[Out]

Tanh[a + b*x]^3/(3*b)

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 42, normalized size = 2.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{\sinh \left ( bx+a \right ) }{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}}+{\frac{\tanh \left ( bx+a \right ) }{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/2*sinh(b*x+a)/cosh(b*x+a)^3+1/2*(2/3+1/3*sech(b*x+a)^2)*tanh(b*x+a))

________________________________________________________________________________________

Maxima [A]  time = 1.0373, size = 18, normalized size = 1.2 \begin{align*} \frac{\tanh \left (b x + a\right )^{3}}{3 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tanh(b*x + a)^3/b

________________________________________________________________________________________

Fricas [B]  time = 1.86072, size = 378, normalized size = 25.2 \begin{align*} -\frac{8 \,{\left (\cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*(cosh(b*x + a)^2 + cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*
sinh(b*x + a)^3 + b*sinh(b*x + a)^4 + 4*b*cosh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^2 + 4*
(b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + 3*b)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh ^{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*tanh(b*x+a)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.21848, size = 42, normalized size = 2.8 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*e^(4*b*x + 4*a) + 1)/(b*(e^(2*b*x + 2*a) + 1)^3)