### 3.982 $$\int \text{sech}^2(x) (a+b \tanh (x))^n \, dx$$

Optimal. Leaf size=19 $\frac{(a+b \tanh (x))^{n+1}}{b (n+1)}$

[Out]

(a + b*Tanh[x])^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0432321, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {3506, 32} $\frac{(a+b \tanh (x))^{n+1}}{b (n+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2*(a + b*Tanh[x])^n,x]

[Out]

(a + b*Tanh[x])^(1 + n)/(b*(1 + n))

Rule 3506

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \text{sech}^2(x) (a+b \tanh (x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^n \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac{(a+b \tanh (x))^{1+n}}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.156339, size = 18, normalized size = 0.95 $\frac{(a+b \tanh (x))^{n+1}}{b n+b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a + b*Tanh[x])^(1 + n)/(b + b*n)

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Maple [A]  time = 0.018, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( a+b\tanh \left ( x \right ) \right ) ^{n+1}}{b \left ( n+1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a+b*tanh(x))^(n+1)/b/(n+1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.44807, size = 220, normalized size = 11.58 \begin{align*} \frac{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \cosh \left (n \log \left (\frac{a \cosh \left (x\right ) + b \sinh \left (x\right )}{\cosh \left (x\right )}\right )\right ) +{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sinh \left (n \log \left (\frac{a \cosh \left (x\right ) + b \sinh \left (x\right )}{\cosh \left (x\right )}\right )\right )}{{\left (b n + b\right )} \cosh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*cosh(x) + b*sinh(x))*cosh(n*log((a*cosh(x) + b*sinh(x))/cosh(x))) + (a*cosh(x) + b*sinh(x))*sinh(n*log((a*
cosh(x) + b*sinh(x))/cosh(x))))/((b*n + b)*cosh(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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