\(\int x^m \cosh (a+b x) \coth (a+b x) \, dx\) [404]

Optimal result

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

[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*csch(b*
x+a),x)

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int x^m \cosh (a+b x) \coth (a+b x) \, dx \]

[In]

Int[x^m*Cosh[a + b*x]*Coth[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*C
sch[a + b*x], x]

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

Mathematica [N/A]

Not integrable

Time = 17.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int x^m \cosh (a+b x) \coth (a+b x) \, dx \]

[In]

Integrate[x^m*Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

Integrate[x^m*Cosh[a + b*x]*Coth[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integral(x^m*cosh(b*x + a)^2*csch(b*x + a), x)

Sympy [F(-1)]

Timed out. \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\text {Timed out} \]

[In]

integrate(x**m*cosh(b*x+a)**2*csch(b*x+a),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate(x^m*cosh(b*x + a)^2*csch(b*x + a), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate(x^m*cosh(b*x + a)^2*csch(b*x + a), x)

Mupad [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int x^m \cosh (a+b x) \coth (a+b x) \, dx=\int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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