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

Optimal. Leaf size=41 \[ \text {Int}\left (\frac {\text {csch}(a+b x)}{x^2},x\right )+b \cosh (a) \text {Chi}(b x)+b \sinh (a) \text {Shi}(b x)-\frac {\sinh (a+b x)}{x} \]

[Out]

b*Chi(b*x)*cosh(a)+b*Shi(b*x)*sinh(a)-sinh(b*x+a)/x+Unintegrable(csch(b*x+a)/x^2,x)

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

Verification is Not applicable to the result.

[In]

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

[Out]

b*Cosh[a]*CoshIntegral[b*x] - Sinh[a + b*x]/x + b*Sinh[a]*SinhIntegral[b*x] + Defer[Int][Csch[a + b*x]/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 38.16, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cosh ^{2}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{x^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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