### 3.86 $$\int \text{sech}^2(c+d x) (a+b \tanh ^2(c+d x)) \, dx$$

Optimal. Leaf size=28 $\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}$

[Out]

(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0293381, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {3675} $\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]

[Out]

(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)

Rule 3675

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

Mathematica [A]  time = 0.0112985, size = 28, normalized size = 1. $\frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)

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Maple [A]  time = 0.036, size = 53, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a\tanh \left ( dx+c \right ) +b \left ( -{\frac{\sinh \left ( dx+c \right ) }{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{\tanh \left ( dx+c \right ) }{2} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) } \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13298, size = 46, normalized size = 1.64 \begin{align*} \frac{b \tanh \left (d x + c\right )^{3}}{3 \, d} + \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.81801, size = 424, normalized size = 15.14 \begin{align*} -\frac{4 \,{\left ({\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (3 \, a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 3 \, a\right )}}{3 \,{\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1891, size = 80, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + b\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*a*e^(4*d*x + 4*c) + 3*b*e^(4*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) + 3*a + b)/(d*(e^(2*d*x + 2*c) + 1)^3)