### 3.369 $$\int x^m \text{sech}(a+b x) \tanh ^2(a+b x) \, dx$$

Optimal. Leaf size=29 $\text{Unintegrable}\left (x^m \text{sech}(a+b x),x\right )-\text{Unintegrable}\left (x^m \text{sech}^3(a+b x),x\right )$

[Out]

Unintegrable[x^m*Sech[a + b*x], x] - Unintegrable[x^m*Sech[a + b*x]^3, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0668012, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int x^m \text{sech}(a+b x) \tanh ^2(a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

Defer[Int][x^m*Sech[a + b*x], x] - Defer[Int][x^m*Sech[a + b*x]^3, x]

Rubi steps

\begin{align*} \int x^m \text{sech}(a+b x) \tanh ^2(a+b x) \, dx &=\int x^m \text{sech}(a+b x) \, dx-\int x^m \text{sech}^3(a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 15.3298, size = 0, normalized size = 0. $\int x^m \text{sech}(a+b x) \tanh ^2(a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\rm sech} \left (bx+a\right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sech(b*x+a)^3*sinh(b*x+a)^2,x)

[Out]

int(x^m*sech(b*x+a)^3*sinh(b*x+a)^2,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(x^m*sech(b*x + a)^3*sinh(b*x + a)^2, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^m*sech(b*x + a)^3*sinh(b*x + a)^2, x)

________________________________________________________________________________________

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(x**m*sech(b*x+a)**3*sinh(b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^m*sech(b*x + a)^3*sinh(b*x + a)^2, x)