### 3.382 $$\int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx$$

Optimal. Leaf size=53 $-\text{Unintegrable}\left (\frac{\tanh (a+b x)}{x^2},x\right )+b \cosh (2 a) \text{Chi}(2 b x)+b \sinh (2 a) \text{Shi}(2 b x)-\frac{\sinh (2 a+2 b x)}{2 x}$

[Out]

b*Cosh[2*a]*CoshIntegral[2*b*x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x] - Unintegrable[Tan
h[a + b*x]/x^2, x]

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Rubi [A]  time = 0.131459, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 19.689, size = 0, normalized size = 0. $\int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm sech} \left (bx+a\right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x\right ) + \frac{1}{2} \, b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x\right ) + \frac{1}{x} + 2 \, \int \frac{1}{x^{2} e^{\left (2 \, b x + 2 \, a\right )} + x^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*e^(-2*a)*gamma(-1, 2*b*x) + 1/2*b*e^(2*a)*gamma(-1, -2*b*x) + 1/x + 2*integrate(1/(x^2*e^(2*b*x + 2*a) +
x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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