\(\int x^m \coth ^2(a+b x) \, dx\) [431]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int x^m \coth ^2(a+b x) \, dx=\text {Int}\left (x^m \coth ^2(a+b x),x\right ) \]

[Out]

Unintegrable(x^m*coth(b*x+a)^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, 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 = {} \[ \int x^m \coth ^2(a+b x) \, dx=\int x^m \coth ^2(a+b x) \, dx \]

[In]

Int[x^m*Coth[a + b*x]^2,x]

[Out]

Defer[Int][x^m*Coth[a + b*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \coth ^2(a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.61 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int x^m \coth ^2(a+b x) \, dx=\int x^m \coth ^2(a+b x) \, dx \]

[In]

Integrate[x^m*Coth[a + b*x]^2,x]

[Out]

Integrate[x^m*Coth[a + b*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int x^m \coth ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^m \coth ^2(a+b x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 145, normalized size of antiderivative = 12.08 \[ \int x^m \coth ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

x*e^(4*b*x + m*log(x) + 4*a)/((m + 1)*e^(4*b*x + 4*a) - 2*(m + 1)*e^(2*b*x + 2*a) + m + 1) + integrate((2*(2*b
*x*e^(4*a) + (m + 1)*e^(4*a))*e^(4*b*x) - (m + 1)*e^(2*b*x + 2*a) - m - 1)*x^m/((m + 1)*e^(6*b*x + 6*a) - 3*(m
+ 1)*e^(4*b*x + 4*a) + 3*(m + 1)*e^(2*b*x + 2*a) - m - 1), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int x^m \coth ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83 \[ \int x^m \coth ^2(a+b x) \, dx=\int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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