3.5.43 \(\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx\) [443]

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

[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 = {} \begin {gather*} \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \end {gather*}

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 = 15.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \end {gather*}

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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Maple [A]
time = 2.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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