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

Optimal. Leaf size=49 \[ -\text {Int}\left (\frac {\tanh (a+b x) \text {sech}(a+b x)}{x^2},x\right )+b \cosh (a) \text {Chi}(b x)+b \sinh (a) \text {Shi}(b x)-\frac {\sinh (a+b x)}{x} \]

[Out]

-CannotIntegrate(sech(b*x+a)*tanh(b*x+a)/x^2,x)+b*Chi(b*x)*cosh(a)+b*Shi(b*x)*sinh(a)-sinh(b*x+a)/x

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 10.23, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right )^{2} \left (\sinh ^{3}\left (b x +a \right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + \frac {1}{2} \, b e^{a} \Gamma \left (-1, -b x\right ) + \frac {2 \, e^{\left (b x + a\right )}}{b x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b x^{2}} + 4 \, \int \frac {e^{\left (b x + a\right )}}{b x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*e^(-a)*gamma(-1, b*x) + 1/2*b*e^a*gamma(-1, -b*x) + 2*e^(b*x + a)/(b*x^2*e^(2*b*x + 2*a) + b*x^2) + 4*in
tegrate(e^(b*x + a)/(b*x^3*e^(2*b*x + 2*a) + b*x^3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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