3.443 \(\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx\)

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\coth (a+b x) \text {csch}(a+b x)}{x},x\right )+\cosh (a) \text {Chi}(b x)+\sinh (a) \text {Shi}(b x) \]

[Out]

CannotIntegrate(coth(b*x+a)*csch(b*x+a)/x,x)+Chi(b*x)*cosh(a)+Shi(b*x)*sinh(a)

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Rubi [A]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Cosh[a]*CoshIntegral[b*x] + Sinh[a]*SinhIntegral[b*x] + Defer[Int][(Coth[a + b*x]*Csch[a + b*x])/x, x]

Rubi steps

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

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Mathematica [A]  time = 24.25, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + \frac {1}{2} \, {\rm Ei}\left (b x\right ) e^{a} - \frac {2 \, e^{\left (b x + a\right )}}{b x e^{\left (2 \, b x + 2 \, a\right )} - b x} - \int \frac {1}{b x^{2} e^{\left (b x + a\right )} + b x^{2}}\,{d x} - \int \frac {1}{b x^{2} e^{\left (b x + a\right )} - b x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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