\(\int x^m \sinh (a+b x) \tanh (a+b x) \, dx\) [355]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}-\text {Int}\left (x^m \text {sech}(a+b x),x\right ) \]

[Out]

1/2*exp(a)*x^m*GAMMA(1+m,-b*x)/b/((-b*x)^m)-1/2*x^m*GAMMA(1+m,b*x)/b/exp(a)/((b*x)^m)-Unintegrable(x^m*sech(b*
x+a),x)

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int x^m \sinh (a+b x) \tanh (a+b x) \, dx \]

[In]

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

[Out]

(E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) - (x^m*Gamma[1 + m, b*x])/(2*b*E^a*(b*x)^m) - Defer[Int][x^m*S
ech[a + b*x], x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \cosh (a+b x) \, dx-\int x^m \text {sech}(a+b x) \, dx \\ & = \frac {1}{2} \int e^{-i (i a+i b x)} x^m \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^m \, dx-\int x^m \text {sech}(a+b x) \, dx \\ & = \frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}-\int x^m \text {sech}(a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.71 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int x^m \sinh (a+b x) \tanh (a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 38.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int x^{m} \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int { x^{m} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int { x^{m} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int x^m \sinh (a+b x) \tanh (a+b x) \, dx=\int \frac {x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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