3.360 \(\int \frac {\sinh (a+b x) \tanh (a+b x)}{x} \, dx\)

Optimal. Leaf size=30 \[ -\text {Int}\left (\frac {\text {sech}(a+b x)}{x},x\right )+\cosh (a) \text {Chi}(b x)+\sinh (a) \text {Shi}(b x) \]

[Out]

Chi(b*x)*cosh(a)+Shi(b*x)*sinh(a)-Unintegrable(sech(b*x+a)/x,x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sinh[a + b*x]*Tanh[a + b*x])/x,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 9.15, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x,x]

[Out]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x, x]

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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