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

Optimal. Leaf size=42 \[ -\text {Int}\left (\frac {\tanh (a+b x)}{x},x\right )+\frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]

[Out]

1/2*cosh(2*a)*Shi(2*b*x)+1/2*Chi(2*b*x)*sinh(2*a)-Unintegrable(tanh(b*x+a)/x,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 = {} \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 19.59, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 2 \, \int \frac {1}{x e^{\left (2 \, b x + 2 \, a\right )} + x}\,{d x} - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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