3.5.16 \(\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx\) [416]

Optimal. Leaf size=40 \[ \frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\text {Int}\left (\frac {\coth (a+b x)}{x},x\right ) \]

[Out]

1/2*cosh(2*a)*Shi(2*b*x)+1/2*Chi(2*b*x)*sinh(2*a)+Unintegrable(coth(b*x+a)/x,x)

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Rubi [A]
time = 0.08, 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 ^2(a+b x) \coth (a+b x)}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a])/2 + (Cosh[2*a]*SinhIntegral[2*b*x])/2 + Defer[Int][Coth[a + b*x]/x, x]

Rubi steps

\begin {align*} \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx &=\int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx\\ &=\int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx\\ &=\frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx\\ &=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 7.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 2.69, size = 0, normalized size = 0.00 \[\int \frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/4*Ei(2*b*x)*e^(2*a) - 1/4*Ei(-2*b*x)*e^(-2*a) - integrate(1/(x*e^(b*x + a) + x), x) + integrate(1/(x*e^(b*x
+ a) - x), x) + log(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)/x,x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^3*csch(b*x + a)/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}{\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)/x,x)

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,\mathrm {sinh}\left (a+b\,x\right )} \,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)),x)

[Out]

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

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