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

Optimal. Leaf size=31 \[ \text {Int}\left (x^m \text {sech}(a+b x),x\right )-\text {Int}\left (x^m \text {sech}^3(a+b x),x\right ) \]

[Out]

Unintegrable(x^m*sech(b*x+a),x)-Unintegrable(x^m*sech(b*x+a)^3,x)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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 = 58.51, size = 0, normalized size = 0.00 \[ \int x^m \text {sech}(a+b x) \tanh ^2(a+b x) \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}, x\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.30, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {sech}\left (b x +a \right )^{3} \left (\sinh ^{2}\left (b x +a \right )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {sech}\left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \]

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________