### 3.505 $$\int \frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x} \, dx$$

Optimal. Leaf size=22 $\text{CannotIntegrate}\left (\frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x},x\right )$

[Out]

CannotIntegrate[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x, x]

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Rubi [A]  time = 0.239708, 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 = {} $\int \frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x,x]

[Out]

Defer[Int][(Csch[a + b*x]^2*Sech[a + b*x]^3)/x, x]

Rubi steps

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

Mathematica [A]  time = 40.4613, size = 0, normalized size = 0. $\int \frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x,x]

[Out]

Integrate[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x, x]

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Maple [A]  time = 0.385, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{2} \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}}{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, b x e^{\left (3 \, b x + 3 \, a\right )} +{\left (3 \, b x e^{\left (5 \, a\right )} - e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )} +{\left (3 \, b x e^{a} + e^{a}\right )} e^{\left (b x\right )}}{b^{2} x^{2} e^{\left (6 \, b x + 6 \, a\right )} + b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2} x^{2}} - 32 \, \int \frac{{\left (3 \, b^{2} x^{2} e^{a} - 2 \, e^{a}\right )} e^{\left (b x\right )}}{32 \,{\left (b^{2} x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{3}\right )}}\,{d x} - 32 \, \int \frac{1}{32 \,{\left (b x^{2} e^{\left (b x + a\right )} + b x^{2}\right )}}\,{d x} - 32 \, \int \frac{1}{32 \,{\left (b x^{2} e^{\left (b x + a\right )} - b x^{2}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*x*e^(3*b*x + 3*a) + (3*b*x*e^(5*a) - e^(5*a))*e^(5*b*x) + (3*b*x*e^a + e^a)*e^(b*x))/(b^2*x^2*e^(6*b*x +
6*a) + b^2*x^2*e^(4*b*x + 4*a) - b^2*x^2*e^(2*b*x + 2*a) - b^2*x^2) - 32*integrate(1/32*(3*b^2*x^2*e^a - 2*e^
a)*e^(b*x)/(b^2*x^3*e^(2*b*x + 2*a) + b^2*x^3), x) - 32*integrate(1/32/(b*x^2*e^(b*x + a) + b*x^2), x) - 32*in
tegrate(1/32/(b*x^2*e^(b*x + a) - b*x^2), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(a + b*x)**2*sech(a + b*x)**3/x, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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