\(\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx\) [341]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\text {Int}\left (x^m \text {sech}(a+b x) \tanh (a+b x),x\right ) \]

[Out]

CannotIntegrate(x^m*sech(b*x+a)*tanh(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 48.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

\[\int x^{m} \operatorname {sech}\left (b x +a \right )^{2} \sinh \left (b x +a \right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int { x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 35.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int x^{m} \sinh {\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int { x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int { x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx=\int \frac {x^m\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int((x^m*sinh(a + b*x))/cosh(a + b*x)^2,x)

[Out]

int((x^m*sinh(a + b*x))/cosh(a + b*x)^2, x)