3.388 \(\int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx\)

Optimal. Leaf size=36 \[ -\text {Int}\left (\frac {\tanh (a+b x) \text {sech}(a+b x)}{x},x\right )+\sinh (a) \text {Chi}(b x)+\cosh (a) \text {Shi}(b x) \]

[Out]

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

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sinh[a + b*x]*Tanh[a + b*x]^2)/x,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 12.09, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x]^2)/x,x]

[Out]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x]^2)/x, x]

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fricas [A]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sech(b*x + a)^2*sinh(b*x + a)^3/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sech(b*x + a)^2*sinh(b*x + a)^3/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2*sinh(b*x+a)^3/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) + 2*integrate(e^(b*x + a)/(
b*x^2*e^(2*b*x + 2*a) + b*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**2*sinh(b*x+a)**3/x,x)

[Out]

Integral(sinh(a + b*x)**3*sech(a + b*x)**2/x, x)

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