### 3.457 $$\int \frac{\coth ^2(a+b x) \text{csch}(a+b x)}{x} \, dx$$

Optimal. Leaf size=27 $\text{Unintegrable}\left (\frac{\text{csch}^3(a+b x)}{x},x\right )+\text{Unintegrable}\left (\frac{\text{csch}(a+b x)}{x},x\right )$

[Out]

Unintegrable[Csch[a + b*x]/x, x] + Unintegrable[Csch[a + b*x]^3/x, x]

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Rubi [A]  time = 0.0715894, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\coth ^2(a+b x) \text{csch}(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(Coth[a + b*x]^2*Csch[a + b*x])/x,x]

[Out]

Defer[Int][Csch[a + b*x]/x, x] + Defer[Int][Csch[a + b*x]^3/x, x]

Rubi steps

\begin{align*} \int \frac{\coth ^2(a+b x) \text{csch}(a+b x)}{x} \, dx &=\int \frac{\text{csch}(a+b x)}{x} \, dx+\int \frac{\text{csch}^3(a+b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 52.2792, size = 0, normalized size = 0. $\int \frac{\coth ^2(a+b x) \text{csch}(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(Coth[a + b*x]^2*Csch[a + b*x])/x,x]

[Out]

Integrate[(Coth[a + b*x]^2*Csch[a + b*x])/x, x]

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Maple [A]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}}{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} +{\left (b x e^{a} + e^{a}\right )} e^{\left (b x\right )}}{b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{2}} + 2 \, \int \frac{b^{2} x^{2} + 2}{4 \,{\left (b^{2} x^{3} e^{\left (b x + a\right )} + b^{2} x^{3}\right )}}\,{d x} + 2 \, \int \frac{b^{2} x^{2} + 2}{4 \,{\left (b^{2} x^{3} e^{\left (b x + a\right )} - b^{2} x^{3}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^3/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )^{3}}{x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^3/x,x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^2*csch(b*x + a)^3/x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**2*csch(b*x+a)**3/x,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )^{3}}{x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*csch(b*x+a)^3/x,x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^2*csch(b*x + a)^3/x, x)