\(\int x^m \coth (a+b x) \, dx\) [397]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int x^m \coth (a+b x) \, dx=\text {Int}\left (x^m \coth (a+b x),x\right ) \]

[Out]

Unintegrable(x^m*coth(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 10, 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 = {} \[ \int x^m \coth (a+b x) \, dx=\int x^m \coth (a+b x) \, dx \]

[In]

Int[x^m*Coth[a + b*x],x]

[Out]

Defer[Int][x^m*Coth[a + b*x], x]

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

Mathematica [N/A]

Not integrable

Time = 6.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \coth (a+b x) \, dx=\int x^m \coth (a+b x) \, dx \]

[In]

Integrate[x^m*Coth[a + b*x],x]

[Out]

Integrate[x^m*Coth[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int x^m \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 108.55 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.70 \[ \int x^m \coth (a+b x) \, dx=\int x^{m} \cosh {\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(x**m*cosh(a + b*x)*csch(a + b*x), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 102, normalized size of antiderivative = 10.20 \[ \int x^m \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

x*e^(2*b*x + m*log(x) + 2*a)/((m + 1)*e^(2*b*x + 2*a) - m - 1) + integrate(((2*b*x*e^(2*a) + (m + 1)*e^(2*a))*
e^(2*b*x) - m - 1)*x^m/((m + 1)*e^(4*b*x + 4*a) - 2*(m + 1)*e^(2*b*x + 2*a) + m + 1), x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int x^m \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.00 \[ \int x^m \coth (a+b x) \, dx=\int \frac {x^m\,\mathrm {cosh}\left (a+b\,x\right )}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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